Introduction
With more than two hundred gravitational-wave (GW) events observed by the LIGO–Virgo–KAGRA collaboration [2 , 3 , 4 , 5 ] , and with the space interferometer LISA [6 , 27 ] and the third-generation ground-based detectors [15 , 84 , 87 ] in preparation, GWs have become a probe not only of their sources but also of the matter they cross on their way to us. Gravitational lensing is one of the most important phenomena in which this is manifest: as for light, the masses along the line of sight deflect, delay and focus the waves [52 , 91 ] . Unlike light, however, GWs can have wavelengths comparable to the Schwarzschild radius of the lens. The ratio of the two is measured by the dimensionless frequency w = 4 G M z ω / c 3 w = 4GM_z\omega/c^3 w = 4 G M z ω / c 3 M z M_z M z w ≫ 1 w \gg1 w ≫ 1 [58 , 59 ] . When w w w [13 , 34 , 71 , 72 , 77 , 100 ] . Which lenses fall in this regime depends on the observed frequency. In the band of ground-based detectors, f ∼ 10 f \sim10 f ∼ 10 10 3 H z 10^{3}\,\mathrm{Hz} 1 0 3 Hz w = 1 w=1 w = 1 8 8 8 800 M ⊙ 800\,M_{\odot} 800 M ⊙ f ∼ 10 − 4 f \sim10^{-4} f ∼ 1 0 − 4 10 − 1 H z 10^{-1}\,\mathrm{Hz} 1 0 − 1 Hz 8 × 10 4 8 \times10^{4} 8 × 1 0 4 8 × 10 7 M ⊙ 8 \times10^{7}\,M_{\odot} 8 × 1 0 7 M ⊙ [13 , 34 , 71 , 72 , 77 , 100 ] . Moreover, interferometers record the strain itself, phase included, so that the complex amplification factor F ( f ) F(f) F ( f ) [30 , 40 ] , whereas in optical astronomy only ∣ F ∣ 2 |F|^{2} ∣ F ∣ 2 [64 ] . Wave-optics signatures are expected to be detectable with present and future interferometers [25 , 31 , 46 , 100 ] , can lift degeneracies between source and lens parameters [28 ] and can probe low-mass halos and compact dark objects [76 ] , and the event GW231123 [1 ] has already been analysed as a possible candidate for wave-optics lensing [22 , 24 , 51 , 57 ] .
The standard description of wave-optics lensing was firstly introduce in the pioneering works of Nakamura and Deguchi [71 , 72 ] (see also [64 , 98 , 100 ] ). Starting from the scalar wave equation in the weak field of a static lens, one reduces the equation for the amplification factor to a Schrödinger equation in which the distance from the source plays the role of time and the inverse frequency that of ℏ \hbar ℏ [49 , 67 ] , and for a thin lens it collapses onto the Kirchhoff–Fresnel diffraction integral over the points of a lens plane, each weighted by the arrival time of the path that crosses it [72 , 91 ] . The same integral follows from a particle Hamiltonian built on the dispersion relation [41 ] ; it has a closed form for a point mass [34 , 71 , 79 ] , it is computed numerically for realistic lenses [102 , 105 , 106 ] , and Picard–Lefschetz methods make it tractable at all frequencies [41 , 60 ] . At high frequency only the stationary points of the arrival time survive, and one recovers the rays, Fermat’s principle and the laws of geometric optics.
This derivation rests on a chain of approximations: the eikonal step, which reduces the second-order wave equation to a first-order, one-way evolution; the paraxial geometry , which replaces the sphere of directions by a plane; a thin lens with source and observer in its far field; a weak field; a static configuration, without which no amplification factor exists; and the scalar treatment of a tensor wave. For a lens at cosmological distance from both source and observer they hold extremely well, but they are not universal, and their physical content is not always transparent. The eikonal step in particular is easily confused with geometric optics, also because the word has several meanings in the literature, although, as we show, it is a different and much weaker condition. Several recent works have looked beyond parts of this chain, with corrections beyond the eikonal approximation [18 ] , a scattering-theory view of lensing [20 , 101 ] , three-dimensional wave optics beyond the thin lens [21 , 97 ] , wave optics in modified gravity, where the amplification factor no longer tends to unity at low frequency [47 ] , polarization effects [29 , 32 , 75 ] , rotating lenses [12 ] and the exact wave equation on a Schwarzschild background [81 ] . When the wave crosses many lenses, the single-lens description gives way to a statistical one: the amplitude and phase fluctuations produced by a random distribution of weak lenses have been computed in wave optics [66 , 69 , 99 ] , and the problem has recently been formulated as a Schwinger–Keldysh path integral in which the lens distribution acts as a quenched disorder, giving a single framework for interference, diffraction and the loss of coherence [7 ] . A natural framework in which all these steps can be followed one at a time is the proper-time technique, or worldline formalism. Introduced by Fock, Nambu, Feynman and Schwinger [42 , 45 , 73 , 95 ] , it writes the propagator of a relativistic field as a sum over the trajectories of an associated particle, parametrized by an auxiliary proper time that is integrated over at the end. It is the basis of the worldline formalism of quantum field theory [8 , 9 , 39 , 92 , 96 ] , in which the proper time is the gauge-invariant length of the einbein of the worldline [16 , 17 , 26 , 56 , 83 , 103 ] ; it has been applied to wave propagation in optics and acoustics [44 , 78 , 93 ] , and in its worldline-quantum-field-theory form it has recently become a tool for the gravitational two-body problem [70 ] . In Ref. [14 ] it was brought into GW lensing. The proper time turns the Helmholtz equation exactly into a first-order Schrödinger equation in which the frequency multiplies the whole worldline action and plays the role of 1 / ℏ 1/\hbar 1/ℏ a posteriori , as a stationary point of an exact integral, so that its validity can be judged: stationarity in the proper time alone, with paraxial and thin-lens paths, gives the diffraction integral, and stationarity also in the path gives the rays. Both are semiclassical statements in the strict sense, ℏ = 1 / ω → 0 \hbar= 1/\omega\to0 ℏ = 1/ ω → 0
Our aims are two. The first is to solve the point-mass lens exactly, beyond the diffraction integral. Its proper-time path integral is a Coulomb problem, which we solve with the method of Duru and Kleinert [37 , 38 , 61 ] , a change of clock along the worldline followed by the Kustaanheimo–Stiefel map [63 ] . The result is an exact representation of the Green function, in which the lens-plane variables and the conditions for geometric optics acquire a direct worldline meaning. The second is to apply this machinery to a configuration in which the approximations behind the diffraction integral fail together: the bound lens , a compact binary orbiting a massive black hole [11 , 53 , 62 , 68 , 90 ] , whose repeated lensing has been proposed as a probe of such systems [19 , 36 , 81 , 104 , 108 ] . There the source is close to the lens, sweeps all angles around it, emits a backscattered echo and moves during the lensing delays. We compute its signal exactly and use it as a benchmark for the approximations used in the literature.
The paper is organized as follows. In § 2 we review the standard route to the diffraction integral, its geometric-optics limit, the point-mass solution, the Fresnel scale and the time-domain description in terms of the delay density. In § 3 we introduce the proper-time technique, Eq. (19) , the role of the einbein and of the choice of clock, and the stationarity conditions that lead to the diffraction integral and to geometric optics. In § 4 we solve the point-mass problem exactly in the Duru–Kleinert gauge, analyse its stationary contributions and show that, for an observer at infinity, the lens plane is spanned by the DK time and the fibre angle. In § 6 we discuss which assumptions of the diffraction integral fail and why, including the exact eikonal condition, the non-relativistic meaning of the one-way step and the backward waves, and we introduce the bound lens, computing the exact signal of a static and of a moving source. In § 7 we derive the approximate models of the literature from the exact formula and define the lensing modulation of a periodic source, and in § 8 we measure their errors near conjunction and on the far side of the orbit, estimate the corrections beyond the scalar linearized model, and discuss chirping sources, distinguishability and the astrophysical window. We conclude in § 9 . Technical details are collected in the appendices. We use units G = c = 1 G = c = 1 G = c = 1
Standard wave optics: the diffraction integral
This section reviews the standard route to the diffraction integral [64 , 72 , 76 , 100 , 101 ] . We consider a static lens in the weak-field (Newtonian) metric
d s 2 = − ( 1 + 2 U ( x ) ) d t 2 + ( 1 − 2 U ( x ) ) d x 2 , ∣ U ∣ ≪ 1 , (1) \mathrm{d}s^2 = -(1+2U(\mathbf{x}))\mathrm{d}t^2 + (1-2U(\mathbf{x}))\mathrm{d}\mathbf{x}^2,\qquad|U| \ll1, \tag*{(1)} d s 2 = − ( 1 + 2 U ( x )) d t 2 + ( 1 − 2 U ( x )) d x 2 , ∣ U ∣ ≪ 1 , (1) where U U U [29 ] also in the wave optics regime. Hence one considers a massless scalar field φ \varphi φ □ φ = 0 \Box\varphi= 0 □ φ = 0 (1) . Because the lens is static, frequency is conserved and we may work with one Fourier component, φ = φ ~ ( x ) e − i ω t \varphi= \tilde{\varphi}(\mathbf{x})e^{-i\omega t} φ = φ ~ ( x ) e − iω t U U U
[ ∇ 2 + ω 2 n 2 ( x ) ] φ ~ = 0 , n 2 = 1 − 4 U , (2) [\nabla^2+\omega^2n^2(\mathbf{x})]\tilde{\varphi}=0,\qquad n^2=1-4U, \tag*{(2)} [ ∇ 2 + ω 2 n 2 ( x )] φ ~ = 0 , n 2 = 1 − 4 U , (2) where ∇ 2 \nabla^2 ∇ 2 n = 1 − 2 U > 1 n=1-2U>1 n = 1 − 2 U > 1 (2) are usually written in terms of the amplification factor
F ( x ) ≡ φ ~ ( x ) φ ~ 0 ( x ) . (3) F(\mathbf{x}) \equiv\frac{\tilde{\varphi}(\mathbf{x})}{\tilde{\varphi}_0(\mathbf{x})}. \tag*{(3)} F ( x ) ≡ φ ~ 0 ( x ) φ ~ ( x ) . (3) Because the field equation is linear and time-independent, the complex amplification factor multiplies the unlensed signal in the frequency domain h ~ l e n s e d ( f ) = F ( f ) h ~ ( f ) \tilde{h}_{\rm lensed}(f)=F(f)\tilde{h}(f) h ~ lensed ( f ) = F ( f ) h ~ ( f ) F ( f ) F(f) F ( f ) arg F \arg F arg F h ( t ) h(t) h ( t ) F ( f ) F(f) F ( f ) [30 , 40 ] . By contrast, optical observations typically measure only ∣ F ∣ 2 |F|^2 ∣ F ∣ 2 [64 ] .
Assuming a point source sitting at the origin and using spherical coordinates ( r , θ , ϕ ) (r,\theta,\phi) ( r , θ , ϕ ) φ ~ = F e i ω r / r \tilde{\varphi}=F e^{i\omega r}/r φ ~ = F e iω r / r (2) gives [72 , 100 ]
∂ 2 F ∂ r 2 + 2 i ω ∂ F ∂ r + 1 r 2 ∇ θ 2 F = 4 ω 2 U F , (4) \frac{\partial^2 F}{\partial r^2}+2i\omega\frac{\partial F}{\partial r}+\frac{1}{r^2}\nabla_\theta^2F=4\omega^2UF, \tag*{(4)} ∂ r 2 ∂ 2 F + 2 iω ∂ r ∂ F + r 2 1 ∇ θ 2 F = 4 ω 2 U F , (4) with ∇ θ 2 \nabla_\theta^2 ∇ θ 2 r r r
∣ ∂ 2 F ∂ r 2 ∣ ≪ ∣ 2 i ω ∂ F ∂ r ∣ , (5) \left|\frac{\partial^2F}{\partial r^2}\right|\ll\left|2i\omega\frac{\partial F}{\partial r}\right|, \tag*{(5)} ∂ r 2 ∂ 2 F ≪ 2 iω ∂ r ∂ F , (5) assuming that ω / ∣ ∂ ln F / ∂ r ∣ ∼ \omega/|\partial\ln F/\partial r|\sim ω /∣ ∂ ln F / ∂ r ∣ ∼ F F F ≫ 1 \gg1 ≫ 1 eikonal approximation 1 [14 , 72 , 98 ] .
Physically, it states that the envelope F F F § 6 shows that it also selects the waves moving away from the source and discards those moving back. In addition, it is standardly assumed that the waves stay close to the axis, θ ≪ 1 \theta\ll1 θ ≪ 1 θ = ( θ cos ϕ , θ sin ϕ ) \boldsymbol{\theta}=(\theta\cos\phi,\theta\sin\phi) θ = ( θ cos ϕ , θ sin ϕ ) ∇ θ 2 → ∇ θ 2 \nabla_{\theta}^{2}\to\nabla_{\boldsymbol{\theta}}^{2} ∇ θ 2 → ∇ θ 2 paraxial approximation). With these assumptions, Equation (4) becomes
i ∂ F ∂ r = − 1 2 ω r 2 ∇ θ 2 F + 2 ω U F . (6) \mathrm{i}\frac{\partial F}{\partial r}=-\frac{1}{2\omega r^{2}}\nabla_{\boldsymbol{\theta}}^{2}F+2\omega U F. \tag*{(6)} i ∂ r ∂ F = − 2 ω r 2 1 ∇ θ 2 F + 2 ω U F . (6) This is a Schrödinger equation in which the distance r r r ω \omega ω 1 / ω 1/\omega 1/ ω ℏ \hbar ℏ 2 ω U 2\omega U 2 ω U θ ( r ) \boldsymbol{\theta}(r) θ ( r )
F ( x O ) = ∫ D θ ( r ) exp { i ω ∫ 0 r O d r [ r 2 2 ∣ d θ d r ∣ 2 − 2 U ( r , θ ) ] } . (7) F(\boldsymbol{x}_{O})=\int\mathcal{D}\boldsymbol{\theta}(r)\exp\left\{\mathrm{i}\omega\int_{0}^{r_{O}}\mathrm{d}r\left[\frac{r^{2}}{2}\left|\frac{\mathrm{d}\boldsymbol{\theta}}{\mathrm{d}r}\right|^{2}-2U(r,\boldsymbol{\theta})\right]\right\}. \tag*{(7)} F ( x O ) = ∫ D θ ( r ) exp { i ω ∫ 0 r O d r [ 2 r 2 d r d θ 2 − 2 U ( r , θ ) ] } . (7) If the lens is thin compared with the distances D L D_{L} D L D L S D_{LS} D L S D S D_{S} D S 2 U → δ ( r − D L S ) ψ ^ 2U\to\delta(r-D_{LS})\hat{\psi} 2 U → δ ( r − D L S ) ψ ^ θ \boldsymbol{\theta} θ β \boldsymbol{\beta} β
F ( ω , β ) = ω 2 π i ( 1 + z L ) D L D S c D L S ∫ d 2 θ exp { i ω t d ( θ , β ) } , t d = ( 1 + z L ) D L D S c D L S [ 1 2 ∣ θ − β ∣ 2 − ψ ( θ ) ] . (8) F(\omega,\boldsymbol{\beta})=\frac{\omega}{2\pi\mathrm{i}}\frac{(1+z_{L})D_{L}D_{S}}{cD_{LS}}\int\mathrm{d}^{2}\boldsymbol{\theta}\,\exp\{\mathrm{i}\omega t_{d}(\boldsymbol{\theta},\boldsymbol{\beta})\},\qquad t_{d}=\frac{(1+z_{L})D_{L}D_{S}}{cD_{LS}}\left[\frac{1}{2}\lvert\boldsymbol{\theta}-\boldsymbol{\beta}\rvert^{2}-\psi(\boldsymbol{\theta})\right]. \tag*{(8)} F ( ω , β ) = 2 π i ω c D L S ( 1 + z L ) D L D S ∫ d 2 θ exp { i ω t d ( θ , β )} , t d = c D L S ( 1 + z L ) D L D S [ 2 1 ∣ θ − β ∣ 2 − ψ ( θ ) ] . (8) where the distances are angular-diameter distances, z L z_{L} z L ψ \psi ψ t d t_{d} t d θ \boldsymbol{\theta} θ
For w → ∞ w\to\infty w → ∞ T T T
∇ x T ( x , y ) = 0 ⟺ y = x − ∇ x ψ ( x ) (lens equation) . (9) \nabla_{\boldsymbol{x}}T(\boldsymbol{x},\boldsymbol{y})=0\quad\Longleftrightarrow\quad\boldsymbol{y}=\boldsymbol{x}-\nabla_{\boldsymbol{x}}\psi(\boldsymbol{x})\qquad\text{(lens equation)}. \tag*{(9)} ∇ x T ( x , y ) = 0 ⟺ y = x − ∇ x ψ ( x ) (lens equation) . (9) which is Fermat’s principle: the rays are the paths of stationary arrival time. This is the standard result of Geometric Optics [58 , 59 ] . Expanding T T T x j \boldsymbol{x}_{j} x j
F G O = ∑ j ∣ μ j ∣ 1 / 2 exp ( i w T j − i π n j ) , μ j = [ det ∂ y ∂ x ∣ x j ] − 1 . (10) F_{\mathrm{GO}}=\sum_{j}\lvert\mu_{j}\rvert^{1/2}\exp(\mathrm{i}wT_{j}-\mathrm{i}\pi n_{j}),\qquad\mu_{j}=\left[\det\frac{\partial\boldsymbol{y}}{\partial\boldsymbol{x}}\bigg|_{\boldsymbol{x}_{j}}\right]^{-1}. \tag*{(10)} F GO = j ∑ ∣ μ j ∣ 1/2 exp ( i w T j − i π n j ) , μ j = [ det ∂ x ∂ y x j ] − 1 . (10) with n j = 0 n_{j}=0 n j = 0 1 2 \frac{1}{2} 2 1 1 1 1 T T T t j = 4 M T j t_{j}=4MT_{j} t j = 4 M T j
The point mass
For a point mass, Eq. (2) becomes a Coulomb problem,
( ∇ 2 + ω 2 + κ r ) φ ~ = 0 , κ ≡ 4 M ω 2 = w ω , (11) \left(\nabla^2+\omega^2+\frac{\kappa}{r}\right)\tilde{\varphi}=0,\qquad\kappa\equiv4M\omega^2=w\omega, \tag*{(11)} ( ∇ 2 + ω 2 + r κ ) φ ~ = 0 , κ ≡ 4 M ω 2 = w ω , (11) equivalent to an attractive Coulomb potential with energy-dependent coupling. We define the Sommerfeld parameter ν ≡ w / 2 = 2 M ω \nu\equiv w/2=2M\omega ν ≡ w /2 = 2 M ω ψ = θ E 2 ln ∣ θ ∣ \psi=\theta_{\mathrm{E}}^2\ln|\theta| ψ = θ E 2 ln ∣ θ ∣ θ E 2 = 4 G M D L S / ( c 2 D L D S ) \theta_{\mathrm{E}}^2=4GMD_{\mathrm{LS}}/(c^2D_{\mathrm{L}}D_{\mathrm{S}}) θ E 2 = 4 GM D LS / ( c 2 D L D S ) θ = θ E x \theta=\theta_{\mathrm{E}}x θ = θ E x β = θ E y \beta=\theta_{\mathrm{E}}y β = θ E y t d = ( 4 G M z / c 3 ) T ( x , y ) t_{\mathrm{d}}=(4GM_z/c^3)\mathcal{T}(x,y) t d = ( 4 G M z / c 3 ) T ( x , y )
F ( w , y ) = w 2 π i ∫ d 2 x e i w T ( x , y ) , T ( x , y ) = 1 2 ∣ x − y ∣ 2 − ln ∣ x ∣ . (12) F(w,y)=\frac{w}{2\pi i}\int\mathrm{d}^2x\,e^{iw\mathcal{T}(x,y)},\qquad\mathcal{T}(x,y)=\frac{1}{2}|x-y|^2-\ln|x|. \tag*{(12)} F ( w , y ) = 2 π i w ∫ d 2 x e i w T ( x , y ) , T ( x , y ) = 2 1 ∣ x − y ∣ 2 − ln ∣ x ∣. (12) The prefactor is a normalization that ensures F = 1 F=1 F = 1 ξ = x ξ E \xi=x\xi_{\mathrm{E}} ξ = x ξ E ξ E = θ E D L = ( 4 G M D e f f ) 1 / 2 / c \xi_{\mathrm{E}}=\theta_{\mathrm{E}}D_{\mathrm{L}}=(4GMD_{\mathrm{eff}})^{1/2}/c ξ E = θ E D L = ( 4 GM D eff ) 1/2 / c D e f f = D L D L S / D S D_{\mathrm{eff}}=D_{\mathrm{L}}D_{\mathrm{LS}}/D_{\mathrm{S}} D eff = D L D LS / D S (12) is the Kirchhoff (or Fresnel–Kirchhoff) diffraction integral for a thin phase screen [41 , 91 ] . The diffraction integral can also be evaluated exactly, yielding a confluent hypergeometric function [34 , 71 , 79 ] :
F ( w , y ) = ( − i w 2 ) i w / 2 Γ ( 1 − i w 2 ) 1 F 1 ( i w 2 ; 1 ; i w 2 y 2 ) . (13) F(w,y)=\left(-\frac{iw}{2}\right)^{iw/2}\Gamma\left(1-\frac{iw}{2}\right){}_1F_1\left(\frac{iw}{2};1;\frac{iw}{2}y^2\right). \tag*{(13)} F ( w , y ) = ( − 2 i w ) i w /2 Γ ( 1 − 2 i w ) 1 F 1 ( 2 i w ; 1 ; 2 i w y 2 ) . (13) It is important to notice that in the limit of long wavelengths w → 0 w\to0 w → 0 F → 1 F\to1 F → 1 [64 ] .
When studying the GO limit, one finds that the lens equation has two images on opposite sides of the lens, with radii and signed magnifications
x ± = 1 2 ( y 2 + 4 ± y ) , μ ± = y 2 + 2 2 y y 2 + 4 ± 1 2 , (14) x_{\pm}=\frac{1}{2}\left(\sqrt{y^2+4}\pm y\right),\qquad\mu_{\pm}=\frac{y^2+2}{2y\sqrt{y^2+4}}\pm\frac{1}{2}, \tag*{(14)} x ± = 2 1 ( y 2 + 4 ± y ) , μ ± = 2 y y 2 + 4 y 2 + 2 ± 2 1 , (14) where x + x_+ x + T \mathcal{T} T x − x_- x − [85 ] . Their time-delay difference is
Δ τ ^ = T − − T + = 1 2 y y 2 + 4 + ln y 2 + 4 + y y 2 + 4 − y ≃ 2 y ( y ≪ 1 ) , (15) \Delta\hat{\tau}=\mathcal{T}_{-}-\mathcal{T}_{+}=\frac{1}{2}y\sqrt{y^2+4}+\ln\frac{\sqrt{y^2+4}+y}{\sqrt{y^2+4}-y}\simeq2y\quad(y\ll1), \tag*{(15)} Δ τ ^ = T − − T + = 2 1 y y 2 + 4 + ln y 2 + 4 − y y 2 + 4 + y ≃ 2 y ( y ≪ 1 ) , (15) in units of 4 M 4M 4 M Δ t ≃ 8 M y \Delta t\simeq8My Δ t ≃ 8 M y
F G O = μ + e i w T + − i μ − e i w T − , (16) F_{\mathrm{GO}}=\sqrt{\mu_+}e^{iw\mathcal{T}_+}-i\sqrt{\mu_-}e^{iw\mathcal{T}_-}, \tag*{(16)} F GO = μ + e i w T + − i μ − e i w T − , (16) where − i = e − i π / 2 -i=e^{-i\pi/2} − i = e − iπ /2 w ≫ 1 w\gg1 w ≫ 1 w y ≫ 1 wy\gg1 w y ≫ 1 (13) gives ∣ F ( w , 0 ) ∣ 2 = π w / ( 1 − e − π w ) |F(w,0)|^2=\pi w/(1-e^{-\pi w}) ∣ F ( w , 0 ) ∣ 2 = π w / ( 1 − e − π w ) w w w ( μ + + μ − ) 2 ≃ 2 / y (\sqrt{\mu_+}+\sqrt{\mu_-})^2\simeq2/y ( μ + + μ − ) 2 ≃ 2/ y w y ≳ 1 wy\gtrsim1 w y ≳ 1 Figure 1 (b) shows the error of Eq. (16) , which scales as ≃ 1 / ( 8 w y ) \simeq1/(8wy) ≃ 1/ ( 8 w y ) y y y [98 ] .
Figure 1. Point-lens amplification factor. (a) ∣ F ( w , y ) ∣ |F(w,y)| ∣ F ( w , y ) ∣ (13) (solid) and from geometric optics, Eq. (16) (dashed). For w ≪ 1 w \ll1 w ≪ 1 F F F μ + + μ − \sqrt{\mu_{+}}+\sqrt{\mu_{-}} μ + + μ − 1 / ( 8 w y ) 1/(8wy) 1/ ( 8 w y ) 0.3 ≤ w ≤ 3 0.3 \le w \le3 0.3 ≤ w ≤ 3
Both limits have a simple meaning in terms of the Fresnel scale. A path through a lens-plane point at distance ξ \xi ξ ξ 2 / 2 D e f f \xi^2/2D_{\mathrm{eff}} ξ 2 /2 D eff r F = ( c D e f f / ω ) 1 / 2 r_{\mathrm{F}}=(cD_{\mathrm{eff}}/\omega)^{1/2} r F = ( c D eff / ω ) 1/2 Figure 2 ), and w = ( ξ E / r F ) 2 w=(\xi_{\mathrm{E}}/r_{\mathrm{F}})^2 w = ( ξ E / r F ) 2 w ≪ 1 w\ll1 w ≪ 1 F ≃ 1 F\simeq1 F ≃ 1 w ≫ 1 w \gg1 w ≫ 1 [60 ] . Finally, since F F F
Figure 2. The Fresnel radius. (a) Each lens-plane point defines a detour path, whose extra phase is one radian at ξ = r F \xi=r_{\mathrm{F}} ξ = r F r F r_{\mathrm{F}} r F w = 0.2 w=0.2 w = 0.2 w = 10 w=10 w = 10
ρ ^ ( τ ^ ; y ) = ∫ d 2 x δ ( τ ^ − T ( x , y ) ) , (17) \hat{\rho}(\hat{\tau};y)=\int\mathrm{d}^{2}x\,\delta\left(\hat{\tau}-\mathcal{T}(x,y)\right), \tag*{(17)} ρ ^ ( τ ^ ; y ) = ∫ d 2 x δ ( τ ^ − T ( x , y ) ) , (17) the area of the lens plane per unit Fermat delay [102 , 105 , 106 ] ; its singularities are the images, a step at a minimum and a logarithm at a saddle.
The proper-time (worldline) technique
We saw that the diffraction integral assumes a scalar wave propagating through a weak, static, thin lens, with source and observer in the far field, under the eikonal and paraxial approximations. These two latter approximations are used in order to obtain a Schrödinger-like equation, which solution can be expressed in terms of a particle path integral Eq. (7) .
In order to obtain a first-order evolution equation from Eq. (2) without having to rely on any approximation, one can introduce an auxiliary evolution parameter, the proper time , that is not a coordinate at all. This is Fock’s and Schwinger’s construction [42 , 45 , 73 , 95 ] , the basis of the worldline formalism of quantum field theory [9 , 39 , 92 , 96 ] , and it is exact. It was first brought it into GW lensing in [14 , 21 ] . The price of this approach is an extra integral, over the proper time, which carries much of the physics. Its reward is that every approximation is taken a posteriori, as a saddle point of an exact integral, so its validity can be judged. In this section we do a short review of this formalism applied to the Coulomb problem. We keep the frequency explicit as the inverse of ℏ \hbar ℏ [49 , 67 ] .
Introducing a point source, Eq. (11) can be written as H ^ ω G ω = δ 3 ( x − x ′ ) \hat{H}_{\omega}G_{\omega}=\delta^{3}(x-x') H ^ ω G ω = δ 3 ( x − x ′ ) H ^ ω = − ∇ 2 − ω 2 n 2 \hat{H}_{\omega}=-\nabla^{2}-\omega^{2}n^{2} H ^ ω = − ∇ 2 − ω 2 n 2 ω 2 \omega^{2} ω 2
H ^ ω = ω 2 H ^ , H ^ = p ^ 2 − 1 + V ( x ) , p ^ = − i ω ∇ , V = 1 − n 2 = − 4 M r . (18) \hat{H}_{\omega}=\omega^{2}\hat{H},\qquad\hat{H}=\hat{p}^{2}-1+V(x),\qquad\hat{p}=-\frac{i}{\omega}\nabla,\qquad V=1-n^{2}=-\frac{4M}{r}. \tag*{(18)} H ^ ω = ω 2 H ^ , H ^ = p ^ 2 − 1 + V ( x ) , p ^ = − ω i ∇ , V = 1 − n 2 = − r 4 M . (18) The momentum operator is p ^ = − i ω ∇ \hat{p}=-\frac{i}{\omega}\nabla p ^ = − ω i ∇ V V V H ( x , p ) = p 2 − n 2 ( x ) H(x,p)=p^{2}-n^{2}(x) H ( x , p ) = p 2 − n 2 ( x ) H ^ \hat{H} H ^ ( a − i ϵ ) − 1 = i ω ∫ 0 ∞ d τ e − i ω τ ( a − i ϵ ) (a-i\epsilon)^{-1}=i\omega\int_{0}^{\infty}d\tau\,e^{-i\omega\tau(a-i\epsilon)} ( a − i ϵ ) − 1 = iω ∫ 0 ∞ d τ e − iω τ ( a − i ϵ ) a a a ϵ > 0 \epsilon>0 ϵ > 0 ω > 0 \omega>0 ω > 0 [94 ] , applied to H ^ = H ^ ω / ω 2 \hat{H}=\hat{H}_{\omega}/\omega^{2} H ^ = H ^ ω / ω 2
G ω ( x , x ′ ) = i ω ∫ 0 ∞ d τ e i ω τ ( 1 + i ϵ ) G ~ ω ( x , x ′ ; τ ) , G ~ ω ( x , x ′ ; τ ) = ⟨ x ∣ e − i ω τ ( p ^ 2 + V ) ∣ x ′ ⟩ . (19) G_{\omega}(x,x')=\frac{i}{\omega}\int_{0}^{\infty}d\tau\,e^{i\omega\tau(1+i\epsilon)}\widetilde{G}_{\omega}(x,x';\tau),\qquad\widetilde{G}_{\omega}(x,x';\tau)=\langle x|e^{-i\omega\tau(\hat{p}^{2}+V)}|x'\rangle. \tag*{(19)} G ω ( x , x ′ ) = ω i ∫ 0 ∞ d τ e iω τ ( 1 + i ϵ ) G ω ( x , x ′ ; τ ) , G ω ( x , x ′ ; τ ) = ⟨ x ∣ e − iω τ ( p ^ 2 + V ) ∣ x ′ ⟩ . (19) The proper time τ \tau τ s s s ω 2 \omega^{2} ω 2 e i s ω 2 e^{is\omega^{2}} e i s ω 2 τ = ω s \tau=\omega s τ = ω s [14 ] . The kernel G ~ ω \widetilde{G}_{\omega} G ω
i ω ∂ G ~ ω ∂ τ = ( − 1 ω 2 ∇ 2 + V ) G ~ ω , G ~ ω ( τ = 0 ) = δ 3 ( x − x ′ ) . (20) \frac{i}{\omega}\frac{\partial\widetilde{G}_{\omega}}{\partial\tau}=\left(-\frac{1}{\omega^{2}}\nabla^{2}+V\right)\widetilde{G}_{\omega},\qquad\widetilde{G}_{\omega}(\tau=0)=\delta^{3}(x-x'). \tag*{(20)} ω i ∂ τ ∂ G ω = ( − ω 2 1 ∇ 2 + V ) G ω , G ω ( τ = 0 ) = δ 3 ( x − x ′ ) . (20) a Schrödinger equation in the auxiliary time τ \tau τ ℏ = 1 / ω \hbar=1/\omega ℏ = 1/ ω 1 / 2 1/2 1/2 V V V 1 → 1 + i ϵ 1\to1+i\epsilon 1 → 1 + i ϵ ω 2 → ω 2 + i ϵ \omega^{2}\to\omega^{2}+i\epsilon ω 2 → ω 2 + i ϵ [43 ]
G ~ ω = ∫ D x D p exp { i ω ∫ 0 τ d τ ′ [ p ⋅ x ˙ − ( p 2 + V ) ] } . (21) \widetilde{G}_{\omega}=\int\mathcal{D}x\,\mathcal{D}p\ \exp\left\{i\omega\int_{0}^{\tau}d\tau'\left[p\cdot\dot{x}-(p^{2}+V)\right]\right\}. \tag*{(21)} G ω = ∫ D x D p exp { iω ∫ 0 τ d τ ′ [ p ⋅ x ˙ − ( p 2 + V ) ] } . (21) with x ˙ = d x / d τ ′ \dot{x}=dx/d\tau' x ˙ = d x / d τ ′ τ ′ ∈ [ 0 , τ ] \tau'\in[0,\tau] τ ′ ∈ [ 0 , τ ] τ \tau τ [8 , 16 , 17 , 92 ]
S [ x , p , e ] = ω ∫ 0 1 d ς [ p ⋅ d x d ς − e ( ς ) H ( x , p ) ] , H = p 2 − 1 + V = p 2 − n 2 ( x ) . (22) S[x,p,e]=\omega\int_0^1 d\varsigma\left[\mathbf{p}\cdot\frac{d\mathbf{x}}{d\varsigma}-e(\varsigma)H(\mathbf{x},\mathbf{p})\right],\qquad H=\mathbf{p}^2-1+V=\mathbf{p}^2-n^2(\mathbf{x}). \tag*{(22)} S [ x , p , e ] = ω ∫ 0 1 d ς [ p ⋅ d ς d x − e ( ς ) H ( x , p ) ] , H = p 2 − 1 + V = p 2 − n 2 ( x ) . (22) Although gravitational waves propagate along null worldlines, whose physical proper time vanishes, the parameter τ \tau τ einbein e ( ς ) e(\varsigma) e ( ς ) ς → f ( ς ) \varsigma\to f(\varsigma) ς → f ( ς ) e d ς e\,d\varsigma e d ς ς \varsigma ς H = 0 H=0 H = 0 ∣ p ∣ = n |\mathbf{p}|=n ∣ p ∣ = n ∣ ω p ∣ 2 = ω 2 n 2 |\omega\mathbf{p}|^2=\omega^2n^2 ∣ ω p ∣ 2 = ω 2 n 2 τ = ∫ 0 1 e d ς \tau=\int_0^1e\,d\varsigma τ = ∫ 0 1 e d ς (19) [26 , 55 , 56 , 83 , 103 ] . The gauge choice also determines the worldline clock. For the frequency-independent action S / ω S/\omega S / ω
d x d τ ′ = 2 p , d p d τ ′ = − ∇ V = − 4 M x r 3 . (23) \frac{d\mathbf{x}}{d\tau'}=2\mathbf{p},\qquad\frac{d\mathbf{p}}{d\tau'}=-\nabla V=-\frac{4M\mathbf{x}}{r^3}. \tag*{(23)} d τ ′ d x = 2 p , d τ ′ d p = − ∇ V = − r 3 4 M x . (23) On shell, these coincide with the ray equations in a medium of index n n n λ \lambda λ d x / d λ = p d\mathbf{x}/d\lambda=\mathbf{p} d x / d λ = p ∣ d x / d λ ∣ = n |d\mathbf{x}/d\lambda|=n ∣ d x / d λ ∣ = n τ ′ = λ / 2 \tau'=\lambda/2 τ ′ = λ /2 times were fixed, the total time could not vary and Fermat’s principle could not emerge; for a monochromatic wave one must fix the energy instead, which requires a boundary term. Integrating out the einbein and the momenta at their saddle then gives the reduced action S r = ω ∫ d ℓ n ( x ) S_r=\omega\int d\ell\,n(\mathbf{x}) S r = ω ∫ d ℓ n ( x ) [41 ] . The same result can be obtained by requiring stationary in the proper time only and assuming both the paraxial and thin-lens approximations [14 ] , the paths become straight lines broken at one point of the lens plane, the path integral collapses into the diffraction integral (12) .
Requiring in addition that the action is stationary under variations of the path gives the classical rays.
An exact solution for a point mass: the Duru–Kleinert gauge
For a point mass V = − 4 M / r V=-4M/r V = − 4 M / r (19) , with its path integral written out, reads
G ω ( x , x ′ ) = i ω ∫ 0 ∞ d τ e i ω τ ( 1 + i ϵ ) ∫ x ( 0 ) = x ′ x ( τ ) = x D x exp { i ω ∫ 0 τ d τ ′ [ 1 4 ∣ d x d τ ′ ∣ 2 + 4 M r ] } . (24) G_\omega(\mathbf{x},\mathbf{x}')=\frac{i}{\omega}\int_0^\infty d\tau\,e^{i\omega\tau(1+i\epsilon)}\int_{\mathbf{x}(0)=\mathbf{x}'}^{\mathbf{x}(\tau)=\mathbf{x}}\mathcal{D}\mathbf{x}\,\exp\left\{i\omega\int_0^\tau d\tau'\left[\frac{1}{4}\left|\frac{d\mathbf{x}}{d\tau'}\right|^2+\frac{4M}{r}\right]\right\}. \tag*{(24)} G ω ( x , x ′ ) = ω i ∫ 0 ∞ d τ e iω τ ( 1 + i ϵ ) ∫ x ( 0 ) = x ′ x ( τ ) = x D x exp { iω ∫ 0 τ d τ ′ [ 4 1 d τ ′ d x 2 + r 4 M ] } . (24) The frequency multiplies the whole exponent, and the lens sits inside the path integral as the frequency-independent Coulomb potential 4 M / r 4M/r 4 M / r τ \tau τ 1 / ω 1/\omega 1/ ω [37 , 38 ] solved the path integral of the hydrogen atom, which has the same structure as Eq. (24) , with two changes: a change of the clock along the worldline, and a change of variables from R 3 \mathbb{R}^3 R 3 R 4 \mathbb{R}^4 R 4
The clock is fixed by the gauge of the einbein in the action (22) . The Duru–Kleinert (DK) gauge makes the einbein proportional to the distance from the lens [37 , 38 , 61 ] ,
e ( ς ) = τ ~ r ( x ( ς ) ) , τ ~ = ∫ 0 1 d ς e ( ς ) r ( x ( ς ) ) . (25) e(\varsigma)=\tilde{\tau}\,r(\mathbf{x}(\varsigma)),\qquad\tilde{\tau}=\int_0^1d\varsigma\,\frac{e(\varsigma)}{r(\mathbf{x}(\varsigma))}. \tag*{(25)} e ( ς ) = τ ~ r ( x ( ς )) , τ ~ = ∫ 0 1 d ς r ( x ( ς )) e ( ς ) . (25) and its invariant modulus is t ~ \tilde{t} t ~ e e e d τ ′ = e d ς \mathrm{d}\tau'=e\,\mathrm{d}\varsigma d τ ′ = e d ς d t ~ ′ = e d ς / r \mathrm{d}\tilde{t}'=e\,\mathrm{d}\varsigma/r d t ~ ′ = e d ς / r
d t ~ = d τ ′ r . (26) \mathrm{d}\tilde{t}=\frac{\mathrm{d}\tau'}{r}. \tag*{(26)} d t ~ = r d τ ′ . (26) As shown in Figure 3 , the DK clock runs faster near the lens and slower farther away. It also has a direct physical interpretation: along a classical worldline, d τ ′ = d λ / 2 \mathrm{d}\tau'=\mathrm{d}\lambda/2 d τ ′ = d λ /2 d l = ∣ p ∣ d λ ≃ d λ \mathrm{d}l=|\mathbf{p}|\,\mathrm{d}\lambda\simeq\mathrm{d}\lambda d l = ∣ p ∣ d λ ≃ d λ t ~ = ∫ d τ ′ / r = ∫ d l / ( 2 r ) \tilde{t}=\int\mathrm{d}\tau'/r=\int\mathrm{d}l/(2r) t ~ = ∫ d τ ′ / r = ∫ d l / ( 2 r ) (1) is Δ t S h = 2 M ∫ d l / r \Delta t_{\mathrm{Sh}}=2M\int\mathrm{d}l/r Δ t Sh = 2 M ∫ d l / r
Figure 3. The same straight path, at impact parameter b b b d t ~ ′ = d τ ′ / r \mathrm{d}\tilde{t}'=\mathrm{d}\tau'/r d t ~ ′ = d τ ′ / r x k = b sinh ( k δ ) x_k=b\sinh(k\delta) x k = b sinh ( k δ )
4 M t ~ = 4 M ∫ d l 2 r = Δ t S h , ω t ~ = ω Δ t S h . (27) 4M\tilde{t}=4M\int\frac{\mathrm{d}l}{2r}=\Delta t_{\mathrm{Sh}},\qquad\omega\tilde{t}=\omega\Delta t_{\mathrm{Sh}}. \tag*{(27)} 4 M t ~ = 4 M ∫ 2 r d l = Δ t Sh , ω t ~ = ω Δ t Sh . (27) Thus, t ~ \tilde{t} t ~ 4 M 4M 4 M e i ω t ~ \mathrm{e}^{\mathrm{i}\omega\tilde{t}} e i ω t ~ (30) is the corresponding Shapiro phase. In this gauge the term e H eH eH t ~ r H \tilde{t}\,rH t ~ r H
r H = r p 2 − r − 4 M . (28) rH=r\mathbf{p}^{2}-r-4M. \tag*{(28)} r H = r p 2 − r − 4 M . (28) Multiplying the constraint by r > 0 r>0 r > 0 H = p 2 − 1 − 4 M / r H=\mathbf{p}^{2}-1-4M/r H = p 2 − 1 − 4 M / r 1 1 1 ω 2 \omega^{2} ω 2 4 M 4M 4 M r r r
H ^ − 1 = ( r H ^ ) − 1 r . (29) \hat{H}^{-1}=(r\hat{H})^{-1}r. \tag*{(29)} H ^ − 1 = ( r H ^ ) − 1 r . (29) and applying Schwinger’s formula, ( a − i ϵ ) − 1 = i ω ∫ 0 ∞ d t ~ e − i ω t ~ ( a − i ϵ ) (a-\mathrm{i}\epsilon)^{-1}=\mathrm{i}\omega\int_{0}^{\infty}\mathrm{d}\tilde{t}\,\mathrm{e}^{-\mathrm{i}\omega\tilde{t}(a-\mathrm{i}\epsilon)} ( a − i ϵ ) − 1 = i ω ∫ 0 ∞ d t ~ e − i ω t ~ ( a − i ϵ ) r H ^ = r p ^ 2 − ( 1 + i ϵ ) r − 4 M r\hat{H}=r\hat{\mathbf{p}}^{2}-(1+\mathrm{i}\epsilon)r-4M r H ^ = r p ^ 2 − ( 1 + i ϵ ) r − 4 M H ^ \hat{H} H ^
G ω ( x , x ′ ) = i ω r ′ ∫ 0 ∞ d t ~ e i ω t ~ ∫ x ( 0 ) = x ′ x ( t ~ ) = x D x exp { i ω ∫ 0 t ~ d t ~ ′ [ 1 4 r ∣ d x d t ~ ′ ∣ 2 + r ] } . (30) G_{\omega}(\mathbf{x},\mathbf{x}')=\frac{\mathrm{i}}{\omega}r'\int_{0}^{\infty}\mathrm{d}\tilde{t}\,\mathrm{e}^{\mathrm{i}\omega\tilde{t}}\int_{\mathbf{x}(0)=\mathbf{x}'}^{\mathbf{x}(\tilde{t})=\mathbf{x}}\mathcal{D}\mathbf{x}\,\exp\left\{\mathrm{i}\omega\int_{0}^{\tilde{t}}\mathrm{d}\tilde{t}'\left[\frac{1}{4r}\left|\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}\tilde{t}'}\right|^{2}+r\right]\right\}. \tag*{(30)} G ω ( x , x ′ ) = ω i r ′ ∫ 0 ∞ d t ~ e i ω t ~ ∫ x ( 0 ) = x ′ x ( t ~ ) = x D x exp { i ω ∫ 0 t ~ d t ~ ′ [ 4 r 1 d t ~ ′ d x 2 + r ] } . (30) where w = 4 M ω w = 4M\omega w = 4 M ω r ′ = ∣ x ′ ∣ r' = |\mathbf{x}'| r ′ = ∣ x ′ ∣ r r r (29) acts on the source point. Equivalently, the Green function of r ^ H ^ \hat{r}\hat{H} r ^ H ^ H ^ g = δ 3 ( x − x ′ ) / r ′ \hat{H}g = \delta^{3}(\mathbf{x}-\mathbf{x}')/r' H ^ g = δ 3 ( x − x ′ ) / r ′ r ′ r' r ′ (24) , the weight of the proper-time integral has changed from e i ω τ e^{i\omega\tau} e iω τ e i ω ⋅ 4 M t ~ e^{i\omega\cdot4M\tilde{t}} e iω ⋅ 4 M t ~ d τ ′ = r d t ~ ′ d\tau' = r\,d\tilde{t}' d τ ′ = r d t ~ ′
ω { τ + ∫ 0 τ d τ ′ [ 1 4 ∣ d x d τ ′ ∣ 2 + 4 M r ] } = ω { 4 M t ~ + ∫ 0 t ~ d t ~ ′ [ 1 4 r ∣ d x d t ~ ′ ∣ 2 + r ] } . (31) \omega\left\{\tau+\int_{0}^{\tau}d\tau'\left[\frac{1}{4}\left|\frac{d\mathbf{x}}{d\tau'}\right|^{2}+\frac{4M}{r}\right]\right\}=\omega\left\{4M\tilde{t}+\int_{0}^{\tilde{t}}d\tilde{t}'\left[\frac{1}{4r}\left|\frac{d\mathbf{x}}{d\tilde{t}'}\right|^{2}+r\right]\right\}. \tag*{(31)} ω { τ + ∫ 0 τ d τ ′ [ 4 1 d τ ′ d x 2 + r 4 M ] } = ω { 4 M t ~ + ∫ 0 t ~ d t ~ ′ [ 4 r 1 d t ~ ′ d x 2 + r ] } . (31) We see directly that the Coulomb potential has disappeared from the paths. In its place there is the term r r r 1 / 2 r 1/2r 1/2 r (30) is therefore still not Gaussian, and it is only formal, because with a position-dependent mass the time-sliced path integral depends on how the slicing is done [54 , 61 ] . To tackle these problems, we apply the Kustaanheimo–Stiefel (KS) map [63 ] . We write a point u ∈ R 4 \mathbf{u}\in\mathbb{R}^{4} u ∈ R 4 z = ( u 1 + i u 2 , u 3 + i u 4 ) z=(u_{1}+iu_{2},u_{3}+iu_{4}) z = ( u 1 + i u 2 , u 3 + i u 4 )
x i = z † σ i z ( i = 1 , 2 , 3 ) , (32) x_i=z^\dagger\sigma_i z\qquad(i=1,2,3), \tag*{(32)} x i = z † σ i z ( i = 1 , 2 , 3 ) , (32) with σ i \sigma_i σ i ∑ i ( σ i ) a b ( σ i ) c d = 2 δ a d δ b c − δ a b δ c d \sum_i(\sigma_i)_{ab}(\sigma_i)_{cd}=2\delta_{ad}\delta_{bc}-\delta_{ab}\delta_{cd} ∑ i ( σ i ) ab ( σ i ) c d = 2 δ a d δ b c − δ ab δ c d
r = ∣ x ∣ = z † z = ∣ u ∣ 2 , (33) r=|\mathbf{x}|=z^\dagger z=|\mathbf{u}|^2, \tag*{(33)} r = ∣ x ∣ = z † z = ∣ u ∣ 2 , (33) and expanding the Hermitian matrix z z † zz^\dagger z z † { 1 , σ i } \{\mathbb{1},\sigma_i\} { 1 , σ i } z z † = 1 2 ( r + x ⋅ σ ) zz^\dagger=\frac{1}{2}(r+\mathbf{x}\cdot\boldsymbol{\sigma}) z z † = 2 1 ( r + x ⋅ σ ) x \mathbf{x} x z z † zz^\dagger z z † z z z e i θ z e^{i\theta}z e i θ z R 4 \mathbb{R}^{4} R 4 x \mathbf{x} x x \mathbf{x} x θ \theta θ (30) , acting on fibre-independent functions, is
r H ^ = − r ω 2 ∇ x 2 − r − 4 M = − 1 4 ω 2 ∇ u 2 − ∣ u ∣ 2 − 4 M ≡ h ^ − 4 M , h ^ = 1 4 p ^ u 2 − ∣ u ∣ 2 , p ^ u = − i ω ∇ u . (34) r\hat{H}=-\frac{r}{\omega^2}\nabla_x^2-r-4M=-\frac{1}{4\omega^2}\nabla_u^2-|\mathbf{u}|^2-4M\equiv\hat{h}-4M,\qquad\hat{h}=\frac{1}{4}\hat{p}_u^2-|\mathbf{u}|^2,\qquad\hat{\mathbf{p}}_u=-\frac{i}{\omega}\nabla_u. \tag*{(34)} r H ^ = − ω 2 r ∇ x 2 − r − 4 M = − 4 ω 2 1 ∇ u 2 − ∣ u ∣ 2 − 4 M ≡ h ^ − 4 M , h ^ = 4 1 p ^ u 2 − ∣ u ∣ 2 , p ^ u = − ω i ∇ u . (34) This is a four-dimensional isotropic harmonic oscillator with a negative spring constant, an inverted oscillator, appropriate for unbound, scattering orbits, with ℏ = 1 / ω \hbar=1/\omega ℏ = 1/ ω 4 M 4M 4 M (34) is an operator identity, no term of higher order in 1 / ω 1/\omega 1/ ω (30) . Using Eq. (32) , d x / d t ~ ′ = 2 Re ( z † σ z ˙ ) d\mathbf{x}/d\tilde{t}'=2\operatorname{Re}(z^\dagger\boldsymbol{\sigma}\dot{z}) d x / d t ~ ′ = 2 Re ( z † σ z ˙ ) d / d t ~ ′ d/d\tilde{t}' d / d t ~ ′
∣ d x d t ~ ′ ∣ 2 = 4 r ∣ u ˙ ∣ 2 − J 2 , J = 2 Im ( z † z ˙ ) = 2 ( u 1 u ˙ 2 − u 2 u ˙ 1 + u 3 u ˙ 4 − u 4 u ˙ 3 ) , (35) \left|\frac{d\mathbf{x}}{d\tilde{t}'}\right|^2=4r|\dot{\mathbf{u}}|^2-J^2,\qquad J=2\operatorname{Im}(z^\dagger\dot{z})=2(u_1\dot{u}_2-u_2\dot{u}_1+u_3\dot{u}_4-u_4\dot{u}_3), \tag*{(35)} d t ~ ′ d x 2 = 4 r ∣ u ˙ ∣ 2 − J 2 , J = 2 Im ( z † z ˙ ) = 2 ( u 1 u ˙ 2 − u 2 u ˙ 1 + u 3 u ˙ 4 − u 4 u ˙ 3 ) , (35) where J J J z → e i α z z\to e^{i\alpha}z z → e i α z p u = 2 u ˙ \mathbf{p}_u=2\dot{\mathbf{u}} p u = 2 u ˙ u \mathbf{u} u J = 0 J=0 J = 0 (30) reduces to ∣ d x / d t ~ ′ ∣ 2 / 4 r = ∣ u ˙ ∣ 2 |d\mathbf{x}/d\tilde{t}'|^2/4r=|\dot{\mathbf{u}}|^2 ∣ d x / d t ~ ′ ∣ 2 /4 r = ∣ u ˙ ∣ 2 r = ∣ u ∣ 2 r=|\mathbf{u}|^2 r = ∣ u ∣ 2 ∣ u ˙ ∣ 2 + ∣ u ∣ 2 |\dot{\mathbf{u}}|^2+|\mathbf{u}|^2 ∣ u ˙ ∣ 2 + ∣ u ∣ 2
We rewrite the Green function in the Schwinger representation in the DK time,
g ( u , u ′ ) = i ω ∫ 0 ∞ d t ~ e i ω t ~ ⟨ u ∣ e − i ω t ~ h ^ ∣ u ′ ⟩ , (36) g(\mathbf{u},\mathbf{u}')=\frac{i}{\omega}\int_{0}^{\infty}d\tilde{t}\,e^{i\omega\tilde{t}}\langle\mathbf{u}|e^{-i\omega\tilde{t}\hat{h}}|\mathbf{u}'\rangle, \tag*{(36)} g ( u , u ′ ) = ω i ∫ 0 ∞ d t ~ e iω t ~ ⟨ u ∣ e − iω t ~ h ^ ∣ u ′ ⟩ , (36) and writing the kernel of h ^ \hat{h} h ^
G ω ( x , x ′ ) = i 8 ω ∫ 0 2 π d θ ′ ∫ 0 ∞ d t ~ e i ω t ~ ∫ u ( 0 ) = e i θ ′ u ′ u ( t ~ ) = u D u exp { i ω ∫ 0 t ~ d t ~ ′ [ ∣ d u d t ~ ′ ∣ 2 + ( 1 + i ϵ ) ∣ u ∣ 2 ] } . (37) G_{\omega}(x,x')=\frac{i}{8\omega}\int_{0}^{2\pi}d\theta'\int_{0}^{\infty}d\tilde{t}\,e^{i\omega\tilde{t}}\int_{u(0)=e^{i\theta'}u'}^{u(\tilde{t})=u}\mathcal{D}u\exp\left\{i\omega\int_{0}^{\tilde{t}}d\tilde{t}'\left[\left|\frac{d u}{d\tilde{t}'}\right|^{2}+(1+i\epsilon)|u|^{2}\right]\right\}. \tag*{(37)} G ω ( x , x ′ ) = 8 ω i ∫ 0 2 π d θ ′ ∫ 0 ∞ d t ~ e iω t ~ ∫ u ( 0 ) = e i θ ′ u ′ u ( t ~ ) = u D u exp { iω ∫ 0 t ~ d t ~ ′ [ d t ~ ′ d u 2 + ( 1 + i ϵ ) ∣ u ∣ 2 ] } . (37) This form can be compared directly with the Schwinger representation (24) : in both, the frequency multiplies the entire exponent, while the remaining terms are frequency-independent. The path integral in Eq. (37) describes a free inverted oscillator and can be evaluated exactly, yielding the Mehler kernel
∫ u ( 0 ) = u ′ ′ u ( t ~ ) = u D u e i ω ∫ 0 t ~ d t ~ ′ ( u ˙ 2 + u 2 ) = − ω 2 π 2 sinh 2 t ~ e i ω A c l ( t , θ ′ ) . (38) \int_{u(0)=u''}^{u(\tilde{t})=u}\mathcal{D}u\,e^{i\omega\int_{0}^{\tilde{t}}d\tilde{t}'(\dot{u}^{2}+u^{2})}=-\frac{\omega^{2}}{\pi^{2}\sinh^{2}\tilde{t}}e^{i\omega A_{\mathrm{cl}}(t,\theta')}. \tag*{(38)} ∫ u ( 0 ) = u ′′ u ( t ~ ) = u D u e iω ∫ 0 t ~ d t ~ ′ ( u ˙ 2 + u 2 ) = − π 2 sinh 2 t ~ ω 2 e iω A cl ( t , θ ′ ) . (38) with
A c l ( t , θ ′ ) = ( r + r ′ ) coth t ~ − 2 u ⋅ u ′ ′ sinh t ~ . (39) A_{\mathrm{cl}}(t,\theta')=(r+r')\coth\tilde{t}-\frac{2u\cdot u''}{\sinh\tilde{t}}. \tag*{(39)} A cl ( t , θ ′ ) = ( r + r ′ ) coth t ~ − sinh t ~ 2 u ⋅ u ′′ . (39) Then, the remaining integral over the initial fibre angle can then be recast in a convenient form. Details are given in Appendix A. The exact propagator now takes the form
G ω ( x , x ′ ) = − i ω 4 π ∫ 0 ∞ d t ~ sinh 2 t ~ e i ω t ~ e i ω ( r + r ′ ) coth t ~ J 0 ( 2 ω ϖ sinh t ~ ) , (40) G_{\omega}(x,x')=-\frac{i\omega}{4\pi}\int_{0}^{\infty}\frac{d\tilde{t}}{\sinh^{2}\tilde{t}}e^{i\omega\tilde{t}}e^{i\omega(r+r')\coth\tilde{t}}J_{0}\left(\frac{2\omega\varpi}{\sinh\tilde{t}}\right), \tag*{(40)} G ω ( x , x ′ ) = − 4 π iω ∫ 0 ∞ sinh 2 t ~ d t ~ e iω t ~ e iω ( r + r ′ ) c o t h t ~ J 0 ( sinh t ~ 2 ω ϖ ) , (40) with J 0 ( q ) = 1 2 π ∫ 0 2 π d θ e i q cos θ J_{0}(q)=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,e^{iq\cos\theta} J 0 ( q ) = 2 π 1 ∫ 0 2 π d θ e i q c o s θ ϖ \varpi ϖ
Analyzing the stationary contributions
After performing the Gaussian integral, Eq. (37) remains with the integrals over t ~ \tilde{t} t ~ θ ′ \theta' θ ′
Ψ ( t ~ , θ ′ ) = ω [ 4 M t ~ + A c l ( t ~ , θ ′ ) ] , (41) \Psi(\tilde{t},\theta')=\omega\left[4M\tilde{t}+A_{\mathrm{cl}}(\tilde{t},\theta')\right], \tag*{(41)} Ψ ( t ~ , θ ′ ) = ω [ 4 M t ~ + A cl ( t ~ , θ ′ ) ] , (41) where A c l A_{\mathrm{cl}} A cl ω \omega ω { t ~ , θ ′ } \{\tilde{t},\theta'\} { t ~ , θ ′ }
∂ A c l ∂ t ~ = − E o s c , ∂ A c l ∂ θ ′ = ± J , (42) \frac{\partial A_{\mathrm{cl}}}{\partial\tilde{t}}=-E_{\mathrm{osc}},\qquad\frac{\partial A_{\mathrm{cl}}}{\partial\theta'}=\pm J, \tag*{(42)} ∂ t ~ ∂ A cl = − E osc , ∂ θ ′ ∂ A cl = ± J , (42) where E o s c = u ˙ 2 − u 2 E_{\mathrm{osc}}=\dot{u}^{2}-u^{2} E osc = u ˙ 2 − u 2 J J J (35) . The two stationarity conditions are therefore
∂ t ~ Ψ = ω ( 4 M − E o s c ) = 0 , (43) \partial_{\tilde{t}}\Psi=\omega\left(4M-E_{\mathrm{osc}}\right)=0, \tag*{(43)} ∂ t ~ Ψ = ω ( 4 M − E osc ) = 0 , (43) the Hamiltonian constraint in the DK gauge, r H = 0 r_H=0 r H = 0
∂ θ ′ Ψ = ± ω J = 0 , (44) \partial_{\theta'}\Psi=\pm\omega J=0, \tag*{(44)} ∂ θ ′ Ψ = ± ω J = 0 , (44) the KS constraint, which states that the orbit does not move along the fibre.
Each pair ( t ~ , θ ′ ) (\tilde t,\theta') ( t ~ , θ ′ ) d τ ′ = r d t ~ ′ \mathrm{d}\tau'=r\,\mathrm{d}\tilde t' d τ ′ = r d t ~ ′ (26) , the projected momentum is p x = 1 2 d x / d τ ′ = ( d x / d t ~ ′ ) / ( 2 r ) \mathbf{p}_x=\frac{1}{2}\mathrm{d}\mathbf{x}/\mathrm{d}\tau'=(\mathrm{d}\mathbf{x}/\mathrm{d}\tilde t')/(2r) p x = 2 1 d x / d τ ′ = ( d x / d t ~ ′ ) / ( 2 r ) (35) with u ˙ 2 = E o s c + r \dot{u}^{2}=E_{\mathrm{osc}}+r u ˙ 2 = E osc + r
∣ p x ∣ 2 = 4 r u ˙ 2 − J 2 4 r 2 = 1 + E o s c r − J 2 4 r 2 . (45) \lvert\mathbf{p}_x\rvert^{2}=\frac{4r\dot{u}^{2}-J^{2}}{4r^{2}}=1+\frac{E_{\mathrm{osc}}}{r}-\frac{J^{2}}{4r^{2}}. \tag*{(45)} ∣ p x ∣ 2 = 4 r 2 4 r u ˙ 2 − J 2 = 1 + r E osc − 4 r 2 J 2 . (45) The rays of geometric optics, i.e. the classical trajectories of the lens, have ∣ p x ∣ 2 = 1 + 4 M / r \lvert\mathbf{p}_x\rvert^{2}=1+4M/r ∣ p x ∣ 2 = 1 + 4 M / r E o s c ≠ 4 M E_{\mathrm{osc}}\ne4M E osc = 4 M J ≠ 0 J\ne0 J = 0 t ~ \tilde t t ~ θ ′ \theta' θ ′ E o s c = 4 M E_{\mathrm{osc}}=4M E osc = 4 M J = 0 J=0 J = 0 [82 , 88 ] and correspond to the two images. The other trial orbits are not discarded: they are the paths through all the other points around the lens, and their interference produces the wave-optics effects. Orbits with a DK time close to the stationary one give the finite radial width of the diffraction pattern, while orbits with J ≠ 0 J\ne0 J = 0 w y ≲ 1 w y\lesssim1 w y ≲ 1 § 4.2 ).
This is the counterpart of the trial paths of the diffraction integral [14 , 20 ] . There every path is on the energy shell but is kinked at the lens plane, and stationarity in the lens-plane point removes the kink; here every path solves the equations of motion, and stationarity in t ~ \tilde t t ~ θ ′ \theta' θ ′ § 4.2 ), so the two pictures give the same images and magnifications.
Observer at infinity
We now place the observer at x = r n ^ \mathbf{x}=r\hat{\mathbf{n}} x = r n ^ r → ∞ r\to\infty r → ∞ x ′ \mathbf{x}' x ′
η ≡ r ′ + n ^ ⋅ x ′ , ζ ≡ ω η , (46) \eta\equiv r'+\hat{\mathbf{n}}\cdot\mathbf{x}',\qquad\zeta\equiv\omega\eta, \tag*{(46)} η ≡ r ′ + n ^ ⋅ x ′ , ζ ≡ ω η , (46) the extra length of the route source → \to → → \to → t ~ = χ + 1 2 ln ( 2 ω r ) \tilde t=\chi+\frac{1}{2}\ln(2\omega r) t ~ = χ + 2 1 ln ( 2 ω r ) (40) becomes
G ω ( r n ^ , x ′ ) → e i ω r + i ν ln 2 ω r 4 π r e − i ω n ^ ⋅ x ′ F ω ( x ′ ) , e − i ζ F ω = − i π ∫ − ∞ ∞ d χ ∫ 0 2 π d θ e − 2 χ e i Ψ ( χ , θ ) , (47) G_{\omega}(r\hat{\mathbf{n}},\mathbf{x}')\to\frac{e^{i\omega r+i\nu\ln2\omega r}}{4\pi r}e^{-i\omega\hat{\mathbf{n}}\cdot\mathbf{x}'}F_{\omega}(\mathbf{x}'),\qquad e^{-i\zeta}F_{\omega}=-\frac{i}{\pi}\int_{-\infty}^{\infty}\mathrm{d}\chi\int_{0}^{2\pi}\mathrm{d}\theta\,e^{-2\chi}e^{i\Psi(\chi,\theta)}, \tag*{(47)} G ω ( r n ^ , x ′ ) → 4 π r e iω r + i ν l n 2 ω r e − iω n ^ ⋅ x ′ F ω ( x ′ ) , e − i ζ F ω = − π i ∫ − ∞ ∞ d χ ∫ 0 2 π d θ e − 2 χ e i Ψ ( χ , θ ) , (47) with
Ψ ( χ , θ ) = w χ + e − 2 χ + 2 ζ e − χ cos θ . (48) \Psi(\chi,\theta)=w\chi+e^{-2\chi}+2\sqrt{\zeta}\,e^{-\chi}\cos\theta. \tag*{(48)} Ψ ( χ , θ ) = w χ + e − 2 χ + 2 ζ e − χ cos θ . (48) The integrals can be done in closed form and give Gordon’s Coulomb wave [34 , 50 , 79 ] ,
F ω ( x ′ ) = N w 1 F 1 ( i ν , 1 ; i ω η ) , N w = e π ν / 2 Γ ( 1 − i ν ) , (49) F_{\omega}(\mathbf{x}')=N_{w}\,{}_1F_1(i\nu,1;i\omega\eta),\qquad N_{w}=e^{\pi\nu/2}\Gamma(1-i\nu), \tag*{(49)} F ω ( x ′ ) = N w 1 F 1 ( i ν , 1 ; iω η ) , N w = e π ν /2 Γ ( 1 − i ν ) , (49) which depends on the source position only through η \eta η
∣ x ∣ = 2 w e − χ , arg x = θ + π , y 2 = 2 ζ w = η 2 M . (50) \lvert\mathbf{x}\rvert=\sqrt{\frac{2}{w}}e^{-\chi},\qquad\arg\mathbf{x}=\theta+\pi,\qquad y^{2}=\frac{2\zeta}{w}=\frac{\eta}{2M}. \tag*{(50)} ∣ x ∣ = w 2 e − χ , arg x = θ + π , y 2 = w 2 ζ = 2 M η . (50) Then, e − 2 χ d χ d θ = w 2 d 2 x e^{-2\chi}\,\mathrm{d}\chi\,\mathrm{d}\theta= \frac{w}{2}\,\mathrm{d}^{2}x e − 2 χ d χ d θ = 2 w d 2 x Ψ = w T ( x , y ) − ζ − ν ln ( w / 2 ) \Psi= w\mathcal{T}(x,y) - \zeta- \nu\ln(w/2) Ψ = w T ( x , y ) − ζ − ν ln ( w /2 ) F ( w , y ) ≡ e i ν ln ( w / 2 ) F ω F(w,y) \equiv e^{i\nu\ln(w/2)}F_{\omega} F ( w , y ) ≡ e i ν l n ( w /2 ) F ω (12) : the radius of the lens plane is the DK time and its azimuth is the fibre angle (Appendix § 12 gives both derivations in full). At fixed x x x w = 4 M ω w = 4M\omega w = 4 M ω 2 ω r / ν = r / M 2\omega r/\nu= r/M 2 ω r / ν = r / M
G ω ( r n ^ , x ′ ) ⟶ e i ω Δ 0 4 π r F ( w , y ) , Δ 0 = r − n ^ ⋅ x ′ + 2 M ln r M . (51) G_{\omega}(r\hat{\boldsymbol{n}},\boldsymbol{x}') \longrightarrow\frac{e^{i\omega\Delta_{0}}}{4\pi r}F(w,y), \qquad\Delta_{0} = r-\hat{\boldsymbol{n}}\cdot\boldsymbol{x}' + 2M\ln\frac{r}{M}. \tag*{(51)} G ω ( r n ^ , x ′ ) ⟶ 4 π r e iω Δ 0 F ( w , y ) , Δ 0 = r − n ^ ⋅ x ′ + 2 M ln M r . (51) No stationary-phase, eikonal, paraxial or thin-lens step has been taken: for a point mass and an observer at infinity the thin-lens integral is exact, provided y y y y 2 = η / 2 M y^{2} = \eta/2M y 2 = η /2 M
In these variables the stationary points of § 4.1 become explicit. With u = e − χ u = e^{-\chi} u = e − χ
∂ θ Ψ = − 2 ζ u sin θ = 0 , ∂ χ Ψ = w − 2 u 2 − 2 ζ u cos θ = 0. (52) \partial_{\theta}\Psi= -2\sqrt{\zeta}\,u\sin\theta= 0, \qquad\partial_{\chi}\Psi= w - 2u^{2} - 2\sqrt{\zeta}\,u\cos\theta= 0. \tag*{(52)} ∂ θ Ψ = − 2 ζ u sin θ = 0 , ∂ χ Ψ = w − 2 u 2 − 2 ζ u cos θ = 0. (52) The fibre condition gives θ ∈ { 0 , π } \theta\in\{0,\pi\} θ ∈ { 0 , π } θ = π \theta= \pi θ = π θ = 0 \theta= 0 θ = 0 x ± x_{\pm} x ± (14) (Figure 4 ).
Figure 4. The lens plane in worldline variables. (a) The distance from the lens is the DK time χ \chi χ φ = θ + π \varphi= \theta+ \pi φ = θ + π T \mathcal{T} T T \mathcal{T} T J = 0 J = 0 J = 0 θ \theta θ J ≠ 0 J \ne0 J = 0
This planarity, however, holds for the rays, not for the wave. At fixed DK time the integral over the fibre angle in Eq. (47) can be done exactly,
1 2 π ∫ 0 2 π d θ e i z cos θ = J 0 ( z ) , z = 2 ζ e − χ = w x y = ω Δ t θ 2 . (53) \frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{d}\theta\,e^{iz\cos\theta} = J_{0}(z), \qquad z = 2\sqrt{\zeta}\,e^{-\chi} = wxy = \frac{\omega\Delta t_{\theta}}{2}. \tag*{(53)} 2 π 1 ∫ 0 2 π d θ e i z c o s θ = J 0 ( z ) , z = 2 ζ e − χ = w x y = 2 ω Δ t θ . (53) where Δ t θ = 8 M x y \Delta t_{\theta} = 8Mxy Δ t θ = 8 M x y x x x J ≠ 0 J \ne0 J = 0 J = 0 J = 0 J = 0 z ≲ 1 z \lesssim1 z ≲ 1 z ≫ 1 z \gg1 z ≫ 1 θ = 0 \theta= 0 θ = 0 π \pi π J 0 ( z ) ≃ ( 2 / π z ) 1 / 2 cos ( z − π / 4 ) J_{0}(z) \simeq(2/\pi z)^{1/2}\cos(z-\pi/4) J 0 ( z ) ≃ ( 2/ π z ) 1/2 cos ( z − π /4 ) 1 / ( 8 z ) 1/(8z) 1/ ( 8 z ) Figure 5 ).
Figure 5. Which clock angles contribute to J 0 ( z ) J_0(z) J 0 ( z ) z = w x y z = wxy z = w x y e − i z cos φ \mathrm{e}^{-\mathrm{i}z\cos\varphi} e − i z c o s φ φ = θ + π \varphi= \theta+ \pi φ = θ + π 20 ∘ 20^\circ 2 0 ∘ z = 0.5 z = 0.5 z = 0.5 z = 30 z = 30 z = 30 0 ∘ 0^\circ 0 ∘ 180 ∘ 180^\circ 18 0 ∘
Geometric optics therefore requires the phase to vary rapidly in both directions: around the circle, with curvature z ± = w x ± y z_{\pm} = wx_{\pm}y z ± = w x ± y w ( 1 + x ± 2 ) w(1+x_{\pm}^{2}) w ( 1 + x ± 2 ) w w w Ψ = w T + const \Psi= w\mathcal{T}+\text{const} Ψ = w T + const T \mathcal{T} T x ± ≃ 1 x_{\pm} \simeq1 x ± ≃ 1
w ≫ 1 and w y ≫ 1. (54) w \gg1 \qquad\text{and}\qquad wy \gg1. \tag*{(54)} w ≫ 1 and w y ≫ 1. (54) Near the axis the error of geometric optics depends essentially on w y wy w y E o s c = 4 M E_{\mathrm{osc}} = 4M E osc = 4 M J = 0 J = 0 J = 0 (16) , with Fermat phases, magnifications given by the Hessian and Morse phases. This is the leading order of the expansion in ℏ = 1 / ω \hbar= 1/\omega ℏ = 1/ ω [20 , 29 , 98 ] .
The bound lens
The bound lens
As discussed in § 2 , GW lensing is usually described by the diffraction integral (12) , which rests on the assumptions collected in Table 1 [71 , 72 , 100 ] .
Table 1. Assumptions behind the diffraction integral, their physical content and when they fail.
For a lens at cosmological distance from both source and observer all of them hold extremely well, which is why the diffraction integral is the standard tool. However it is worth asking when they fail, because this is where the worldline representation becomes necessary. In order to do so, one first has to understand the physical and geometric meaning of each one of them. The least obvious is the eikonal step of Eq. (5) , which reduces the second-order wave equation to a first-order, one-way evolution. To make it easier, it is worth analyzing it for a point lens, in which case it can be tested exactly.
For a plane wave incident on a point lens, the exact solution of Eq. (11) is the Coulomb wave, whose envelope F F F ω ( r − z ) \omega(r-z) ω ( r − z ) z z z ∂ z 2 F \partial_z^2 F ∂ z 2 F 2 i ω ∂ z F 2i\omega\partial_z F 2 iω ∂ z F
ε E ≡ ∣ ∂ z 2 F ∣ ∣ 2 ω ∂ z F ∣ (55) \varepsilon_E \equiv\frac{\lvert\partial_z^2 F\rvert}{\lvert2\omega\partial_z F\rvert} \tag*{(55)} ε E ≡ ∣ 2 ω ∂ z F ∣ ∣ ∂ z 2 F ∣ (55) to be small. Evaluating it with the exact Coulomb wave, at distance d d d θ \theta θ
ε E ≃ λ ˉ d + M d + O ( sin 2 θ 2 ) , λ ˉ = 1 ω . (56) \varepsilon_E \simeq\frac{\bar{\lambda}}{d}+\frac{M}{d}+\mathcal{O}\left(\sin^2\frac{\theta}{2}\right), \qquad\bar{\lambda}=\frac{1}{\omega}. \tag*{(56)} ε E ≃ d λ ˉ + d M + O ( sin 2 2 θ ) , λ ˉ = ω 1 . (56) Each term is a separate condition. The first requires the wavelength to be short compared with the distance from the lens, which is the scale over which the field of a point mass varies. The second, M / d = ∣ U ∣ M/d=\lvert U\rvert M / d = ∣ U ∣ z z z z z z ξ E \xi_E ξ E w ≲ 1 w\lesssim1 w ≲ 1 λ ˉ ≪ ξ E \bar{\lambda}\ll\xi_E λ ˉ ≪ ξ E
f ≫ f E ≡ 1 2 π ξ E . (57) f\gg f_E\equiv\frac{1}{2\pi\xi_E}. \tag*{(57)} f ≫ f E ≡ 2 π ξ E 1 . (57) Figure 6 shows ε E \varepsilon_E ε E w = 1 w=1 w = 1 (56) grow: at w = 1 w=1 w = 1 λ ˉ = 4 M \bar{\lambda}=4M λ ˉ = 4 M ε E ≃ 0.04 \varepsilon_E\simeq0.04 ε E ≃ 0.04 100 M 100M 100 M ε E \varepsilon_E ε E z z z R R R ≃ λ ˉ / R \simeq\bar{\lambda}/R ≃ λ ˉ / R sin 2 ( β / 2 ) \sin^2(\beta/2) sin 2 ( β /2 ) π 2 / 4 − 1 ≃ 1.5 \pi^2/4-1\simeq1.5 π 2 /4 − 1 ≃ 1.5 Figure 6 (b)]. The ratio ε E \varepsilon_E ε E ε E ∼ 1 \varepsilon_E\sim1 ε E ∼ 1 r − z r-z r − z y 2 = η / 2 M y^2=\eta/2M y 2 = η /2 M § 4.2 and § 6.2 ). This is a property of the 1 / r 1/r 1/ r
Figure 6. Where the assumptions behind the diffraction integral fail around a point lens, w = 1 w = 1 w = 1 ε E \varepsilon_E ε E R = 100 M R = 100M R = 100 M ± ξ E \pm\xi_E ± ξ E β \beta β ε E \varepsilon_E ε E R = 100 M R = 100M R = 100 M 10 3 M 10^3M 1 0 3 M
The meaning of the eikonal step is clearest in field-theory language. If the distance along the propagation plays the role of time, the frequency that of a mass and the transverse wavenumber that of a momentum, the Helmholtz equation becomes a Klein–Gordon equation, and the eikonal step is its non-relativistic limit [48 , 78 , 93 ] . Just as the non-relativistic limit discards the antiparticles, the eikonal step keeps only the wave moving forward and discards the one moving backward. A backward wave can arise in two ways. The first is a classical turning point, where a ray itself reverses its direction; this requires a strong deflection, i.e. a path within a few gravitational radii of a very compact object, outside the linearized field. The second is backscattering by diffraction, which needs no ray to turn back: the 1 / r 1/r 1/ r § 6.2 ). Its relative amplitude decreases with the distance of the source from the lens, so it is negligible on a cosmological line of sight but not for a source close to its lens.
A nearby source also violates the static assumption. The amplification factor multiplies the spectrum only if the configuration is invariant under time translations, so that each worldline contributes a delayed copy of the emitted signal. If the source moves during the spread of delays of the contributing worldlines, each of them samples it at a different emission time, position and Doppler factor, and no amplification factor exists. The usual treatment, which evaluates the static amplification factor at the instantaneous position of the source [36 , 108 ] , must then be replaced by the exact time-domain Green function, in which each worldline has its own retarded time. The remaining assumptions are milder: the paraxial projection of the source onto a plane is accurate near the lens axis but wrong by a factor of order one at large angles, the thin-lens corrections are suppressed by the ratio of the Schwarzschild radius to the distances, and the weak-field and scalar approximations fail only for paths within a few gravitational radii of the lens or at large scattering angles.
All these effects are negligible on a cosmological line of sight, but they appear together when the source orbits its lens (Figure 6 ): near conjunction the paths pass close to the lens, on the rest of the orbit the source is seen at large angles and its backscattered echo interferes with the direct wave, and along the whole orbit the source moves during the lensing delays. This bound lens is therefore the natural setting for the exact worldline treatment.
The bound lens: triple systems
Normally a gravitational lens is a mass that happens to sit between a distant source and us, with all three objects billions of light-years apart. In a bound lens, or triple system, the source orbits the lens [36 , 62 , 81 , 104 , 108 ] . A typical case is a binary of small black holes circling a much heavier black hole, for example in a galactic nucleus. Once per orbit the source passes behind the lens as we see it, and its signal is lensed. For the rest of the orbit it is beside or in front of the lens [19 , 52 , 81 , 108 ] .
With this idea in mind, we consider a gravitational-wave source on a circular orbit around a point lens. The lens has mass M M M R R R Ω = ( M / R 3 ) 1 / 2 \Omega= (M/R^3)^{1/2} Ω = ( M / R 3 ) 1/2 v = ( M / R ) 1 / 2 v = (M/R)^{1/2} v = ( M / R ) 1/2 n ^ \hat{n} n ^ i i i Φ = Ω t \Phi= \Omega t Φ = Ω t i = 90 ∘ i = 90^\circ i = 9 0 ∘ Figure 7 ).
Figure 7. Geometry of the bound lens. (a) The orbit on the sky for R = 100 M R = 100M R = 100 M cos i = 0.1 \cos i = 0.1 cos i = 0.1 R R R R cos i R\cos i R cos i R cos i = 10 M R\cos i = 10M R cos i = 10 M y min ≃ 0.5 y_{\min} \simeq0.5 y m i n ≃ 0.5 β \beta β R cos β R\cos\beta R cos β η = R − R cos β \eta= R - R\cos\beta η = R − R cos β y = η / 2 M y = \sqrt{\eta/2M} y = η /2 M β \beta β
This configuration combines all the effects that the diffraction integral leaves out. The source is close to the lens, so the waves reach the observer after being deflected by every angle and part of them is scattered back; and the source moves while the signal crosses the lens. Our goal in this section is to compute the received signal exactly within a well-defined model, and then to use the exact result to identify, worldline by worldline, what each standard approximation keeps and where along the orbit it fails.
For the sake of simplicity, we consider a scalar wave propagating in the first order metric of Eq. (1) with a point mass lens potential, meaning
g 00 = − ( 1 − 2 M / r ) , g i j = ( 1 + 2 M / r ) δ i j . (58) g_{00} = -(1 - 2M/r), \qquad g_{ij} = (1 + 2M/r)\delta_{ij}. \tag*{(58)} g 00 = − ( 1 − 2 M / r ) , g ij = ( 1 + 2 M / r ) δ ij . (58) Moreover, we assume the lens to be static, which requires the mass m m m m ≪ M m \ll M m ≪ M ( m / M ) v (m/M)v ( m / M ) v t − t ′ t - t' t − t ′ 10 2 10^2 1 0 2 10 3 M 10^3M 1 0 3 M [20 ] are not negligible, and waves scattered through large angles probe the strong field near the photon sphere, which requires black-hole perturbation theory [23 , 81 , 89 ] . We estimate these corrections, together with those due to the spin of the wave and to the emission pattern of the source, in § 8.4 .
The choice of a point lens deserves a comment. For a static point lens the worldline and the diffraction-integral descriptions agree: the exact far-field Green function of § 4.2 has the form of the point-lens amplification factor (13) at every position of the source, provided the offset is defined by the excess path, y 2 = η / 2 M y^2 = \eta/2M y 2 = η /2 M 1 / r 1/r 1/ r r − z r-z r − z Figure 6 ). For any other lens, and already for the second-order Schwarzschild field, no such property exists. We use the point lens because it is the case that the Duru–Kleinert gauge solves exactly (§ 4 ), so that the treatment of the bound lens rests on an exact propagator rather than on an approximation. Even for this lens, however, the bound configuration contains effects that the diffraction integral does not consider at all. The amplification factor describes a static configuration, whereas the source moves during the lensing delays. Each worldline then delivers the signal emitted at its own retarded time and with its own Doppler factor: the echo received at a given time left the source earlier than the direct wave received with it and, on a circular orbit, carries no orbital Doppler shift, so that no single amplification factor describes the received wave. The offset must be defined by the excess path along the whole orbit, since the small-angle offset misplaces the echo on the far side and a projected one would make a source in front of the lens appear aligned with it. Finally, the echo itself is a wave scattered backwards by the lens, which the one-way equation excludes by construction and which the closed form contains only through the parabolic symmetry. These are the effects captured by the exact time-domain signal of § 6.3 and missed by the approximations of § 7 .
A static source: direct wave and echo
We begin with a source frozen at a point x ′ \boldsymbol{x}' x ′ n ^ ⋅ x ′ = − R sin i cos Φ \hat{\boldsymbol{n}} \cdot\boldsymbol{x}' = -R \sin i \cos\Phi n ^ ⋅ x ′ = − R sin i cos Φ n ^ ⋅ x ′ < 0 \hat{\boldsymbol{n}} \cdot\boldsymbol{x}' < 0 n ^ ⋅ x ′ < 0 Δ R = − n ^ ⋅ x ′ \Delta_R = -\hat{\boldsymbol{n}} \cdot\boldsymbol{x}' Δ R = − n ^ ⋅ x ′ [68 ] . The second is the angle β \beta β x ′ \boldsymbol{x}' x ′ − n ^ -\hat{\boldsymbol{n}} − n ^ cos β = sin i cos Φ \cos\beta= \sin i \cos\Phi cos β = sin i cos Φ β = 0 \beta= 0 β = 0 β = π \beta= \pi β = π β min ≃ cos i \beta_{\min} \simeq\cos i β m i n ≃ cos i
η = r ′ + n ^ ⋅ x ′ = R ( 1 − cos β ) , (59) \eta= r' + \hat{\boldsymbol{n}} \cdot\boldsymbol{x}' = R(1-\cos\beta), \tag*{(59)} η = r ′ + n ^ ⋅ x ′ = R ( 1 − cos β ) , (59) because the direct wave needs a distance R cos β R\cos\beta R cos β R R R Figure 7 (b)]. This excess path ranges from R ( 1 − sin i ) R(1-\sin i) R ( 1 − sin i ) R ( 1 + sin i ) R(1+\sin i) R ( 1 + sin i )
For an observer at infinity, the exact far-field Green function (Eq. (51) ) separates into four factors,
G ω ( r n ^ , x ′ ) → e i ω r 4 π r e 2 i M ω ln ( r / M ) e − i ω n ^ ⋅ x ′ F ( ω , y ) . (60) G_\omega(r\hat{\boldsymbol{n}},\boldsymbol{x}') \to\frac{e^{i\omega r}}{4\pi r}e^{2iM\omega\ln(r/M)}e^{-i\omega\hat{\boldsymbol{n}}\cdot\boldsymbol{x}'}F(\omega,y). \tag*{(60)} G ω ( r n ^ , x ′ ) → 4 π r e iω r e 2 i M ω l n ( r / M ) e − iω n ^ ⋅ x ′ F ( ω , y ) . (60) The first factor is the free propagation to the observer, the second the Shapiro delay accumulated on the long leg from the lens to the observer, and the third the Rømer phase of the source position. The first three are pure delays, linear in ω \omega ω F ( ω , y ) F(\omega,y) F ( ω , y ) (12) ), whose closed form is Eq. (13) , evaluated at the offset
y = ( η 2 M ) 1 / 2 . (61) y=\left(\frac{\eta}{2M}\right)^{1/2}. \tag*{(61)} y = ( 2 M η ) 1/2 . (61) The amplification factor therefore depends on the position of the source only through η = r ′ + n ^ ⋅ x ′ \eta=r^{\prime}+\hat{\boldsymbol{n}}\cdot\boldsymbol{x}^{\prime} η = r ′ + n ^ ⋅ x ′ (2) is symmetric, and the propagator is unchanged when source and observer are exchanged, G ω ( x , x ′ ) = G ω ( x ′ , x ) G_\omega(\boldsymbol{x},\boldsymbol{x}^{\prime})=G_\omega(\boldsymbol{x}^{\prime},\boldsymbol{x}) G ω ( x , x ′ ) = G ω ( x ′ , x ) n ^ \hat{\boldsymbol{n}} n ^ − n ^ -\hat{\boldsymbol{n}} − n ^ F F F e i ω z 1 F 1 ( i ν , 1 ; i ω ( r − z ) ) e^{i\omega z}{}_1F_1(i\nu,1;i\omega(r-z)) e iω z 1 F 1 ( i ν , 1 ; iω ( r − z )) z = − n ^ ⋅ x z=-\hat{\boldsymbol{n}}\cdot\boldsymbol{x} z = − n ^ ⋅ x [50 ] . Its envelope depends on position only through the parabolic coordinate r − z r-z r − z η \eta η F F F y y y η ≃ R β 2 / 2 \eta\simeq R\beta^2/2 η ≃ R β 2 /2 y ≃ β R / ξ E y\simeq\beta R/\xi_E y ≃ β R / ξ E ξ E = ( 4 M R ) 1 / 2 \xi_E=(4MR)^{1/2} ξ E = ( 4 M R ) 1/2 R R R y min ≃ ( R / 4 M ) 1 / 2 ∣ cos i ∣ y_{\min}\simeq(R/4M)^{1/2}|\cos i| y m i n ≃ ( R /4 M ) 1/2 ∣ cos i ∣ R = 100 M R=100M R = 100 M cos i = 0.1 \cos i=0.1 cos i = 0.1 conjunction window the part of the orbit with y < 3 y<3 y < 3 R = 100 M R=100M R = 100 M R = 10 3 M R=10^3M R = 1 0 3 M far side the part with β > π / 2 \beta>\pi/2 β > π /2
Away from conjunction the excess path is many wavelengths, ζ ≡ ω η ≫ 1 + ν 2 \zeta\equiv\omega\eta\gg1+\nu^2 ζ ≡ ω η ≫ 1 + ν 2 ν = w / 2 \nu=w/2 ν = w /2 F F F
F ≃ y − i w [ 1 − i ν 2 ζ + ⋯ ] + ν i ν N w e i ζ ( i ζ ) i ν − 1 Γ ( i ν ) [ 1 + O ( ζ − 1 ) ] . (62) F\simeq y^{-iw}\left[1-\frac{i\nu^2}{\zeta}+\cdots\right]+\nu^{i\nu}N_w\frac{e^{i\zeta}(i\zeta)^{i\nu-1}}{\Gamma(i\nu)}\left[1+O(\zeta^{-1})\right]. \tag*{(62)} F ≃ y − i w [ 1 − ζ i ν 2 + ⋯ ] + ν i ν N w Γ ( i ν ) e i ζ ( i ζ ) i ν − 1 [ 1 + O ( ζ − 1 ) ] . (62) with N w N_w N w (49) . The direct wave carries the Shapiro phase − w ln y -w\ln y − w ln y − ν 2 / ζ -\nu^2/\zeta − ν 2 / ζ echo , has relative amplitude ∣ F s c ∣ = ν / ζ = 2 M / η |F_{\rm sc}|=\nu/\zeta=2M/\eta ∣ F sc ∣ = ν / ζ = 2 M / η e i ζ e^{i\zeta} e i ζ η \eta η [62 ] . Near alignment it becomes the second image. Along the orbit its amplitude ranges from 2 M / R 2M/R 2 M / R M / R M/R M / R R = 100 M R=100M R = 100 M
The static amplification factor along the orbit follows directly. For w = 1 w=1 w = 1 R = 100 M R=100M R = 100 M cos i = 0.1 \cos i=0.1 cos i = 0.1 ∣ F ∣ |F| ∣ F ∣ y = y min = 0.50 y=y_{\min}=0.50 y = y m i n = 0.50 ∣ F ( 1 , 0 ) ∣ = 1.81 |F(1,0)|=1.81 ∣ F ( 1 , 0 ) ∣ = 1.81 μ + + μ − = 2.03 \sqrt{\mu_+}+\sqrt{\mu_-}=2.03 μ + + μ − = 2.03 y = 0.5 y=0.5 y = 0.5 ∣ F ∣ |F| ∣ F ∣ 1 ± 2 M / η 1\pm2M/\eta 1 ± 2 M / η
The worldline picture of § 4.2 gives a meaning to each part of this result. Near conjunction the lens-plane variable x \boldsymbol{x} x ξ E \xi_E ξ E n ^ \hat{\boldsymbol{n}} n ^ ∣ x ∣ |\boldsymbol{x}| ∣ x ∣ arg x \arg\boldsymbol{x} arg x x − ≃ 1 / y ≪ 1 x_-\simeq1/y\ll1 x − ≃ 1/ y ≪ 1 β \beta β b = 2 M cot ( β / 2 ) b=2M\cot(\beta/2) b = 2 M cot ( β /2 ) 2 M 2M 2 M β > π / 2 \beta>\pi/2 β > π /2 § 8.4 ).
In this language, each approximation used in the literature is a stationarity assumption with a definite physical content, and each fails in a definite part of the orbit. Stationarity in the DK time puts every worldline on the energy shell of the lens and fixes the distance of the images from it (the radial lens equation); stationarity in the fibre angle makes every worldline planar and puts the images in the plane of source, lens and observer. Imposing both gives geometric optics. Since y ≃ β ( R / 4 M ) 1 / 2 y \simeq\beta(R/4M)^{1/2} y ≃ β ( R /4 M ) 1/2
β G O ≃ 2 w ( M R ) 1 / 2 (63) \beta_{\mathrm{GO}} \simeq\frac{2}{w}\left(\frac{M}{R}\right)^{1/2} \tag*{(63)} β GO ≃ w 2 ( R M ) 1/2 (63) of conjunction, e.g. 0.2 rad for w = 1 w = 1 w = 1 R = 100 M R = 100M R = 100 M
A simpler approximation retains a single straight worldline and assigns to it the DK phase, corresponding to its Shapiro delay. This is the eikonal approximation in the sense of high-energy scattering [20 ] , and it gives F S L = y − i w F_{\mathrm{SL}} = y^{-iw} F SL = y − i w (62) . Since ∣ F S L ∣ = 1 \lvert F_{\mathrm{SL}}\rvert= 1 ∣ F SL ∣ = 1 ζ ≫ ν ( 1 + ν ) \zeta\gg\nu(1+\nu) ζ ≫ ν ( 1 + ν ) η ≫ 2 M ( 1 + ν ) \eta\gg2M(1+\nu) η ≫ 2 M ( 1 + ν )
β e i k ≃ [ 4 M ( 1 + ν ) R ] 1 / 2 (64) \beta_{\rm eik} \simeq\left[\frac{4M(1+\nu)}{R}\right]^{1/2} \tag*{(64)} β eik ≃ [ R 4 M ( 1 + ν ) ] 1/2 (64) of conjunction (0.24 rad for w = 1 w=1 w = 1 R = 100 M R=100M R = 100 M η \eta η η p a r = R β 2 / 2 \eta_{\rm par}=R\beta^2/2 η par = R β 2 /2 π 2 / 4 \pi^2/4 π 2 /4
A moving source: the exact signal
When the source moves, the amplification factor is no longer defined. The lens, however, is static, so the retarded Green function depends only on t − t ′ t-t' t − t ′ q ( t ) q(t) q ( t ) x s ( t ) \boldsymbol{x}_{s}(t) x s ( t )
φ ( x , t ) = ∫ d t ′ q ( t ′ ) G R ( x , t ; x s ( t ′ ) , t ′ ) (65) \varphi(\boldsymbol{x},t)=\int\mathrm{d}t'\,q(t')G_{\rm R}(\boldsymbol{x},t;\boldsymbol{x}_{s}(t'),t') \tag*{(65)} φ ( x , t ) = ∫ d t ′ q ( t ′ ) G R ( x , t ; x s ( t ′ ) , t ′ ) (65) For an observer at infinity, G R G_{\rm R} G R
G R ( r n ^ , t ; x ′ , t ′ ) = 1 8 π 2 r ∂ ∂ t ρ ^ ( t − t ′ − Δ 0 4 M ; y ) , Δ 0 = r − n ^ ⋅ x ′ + 2 M ln r M (66) G_{\rm R}(r\hat{\boldsymbol{n}},t;\boldsymbol{x}',t')=\frac{1}{8\pi^2r}\frac{\partial}{\partial t}\hat{\rho}\left(\frac{t-t'-\Delta_0}{4M};y\right),\qquad\Delta_0=r-\hat{\boldsymbol{n}}\cdot\boldsymbol{x}'+2M\ln\frac{r}{M} \tag*{(66)} G R ( r n ^ , t ; x ′ , t ′ ) = 8 π 2 r 1 ∂ t ∂ ρ ^ ( 4 M t − t ′ − Δ 0 ; y ) , Δ 0 = r − n ^ ⋅ x ′ + 2 M ln M r (66) in which the lens enters only through the delay density (17) . We remove the delay common to all worldlines by defining the arrival time T ≡ t − r − 2 M ln ( r / M ) T\equiv t-r-2M\ln(r/M) T ≡ t − r − 2 M ln ( r / M ) h ( T ) ≡ 4 π r φ ( r n ^ , t ) h(T)\equiv4\pi r\varphi(r\hat{\boldsymbol{n}},t) h ( T ) ≡ 4 π r φ ( r n ^ , t ) (66) into Eq. (65) and calling now t ′ = t t'=t t ′ = t
h ( T ) = 1 2 π d d T ∫ d t q ( t ) ρ ^ ( T − t − Δ R ( t ) 4 M ; y ( t ) ) (67) h(T)=\frac{1}{2\pi}\frac{\mathrm{d}}{\mathrm{d}T}\int\mathrm{d}t\,q(t)\hat{\rho}\left(\frac{T-t-\Delta_R(t)}{4M};y(t)\right) \tag*{(67)} h ( T ) = 2 π 1 d T d ∫ d t q ( t ) ρ ^ ( 4 M T − t − Δ R ( t ) ; y ( t ) ) (67) where Δ R ( t ) = − n ^ ⋅ x s ( t ) \Delta_R(t)=-\hat{\boldsymbol{n}}\cdot\boldsymbol{x}_{s}(t) Δ R ( t ) = − n ^ ⋅ x s ( t ) y ( t ) y(t) y ( t ) (61) are now evaluated along the orbit. This formula holds for any time dependence of the source, for any orbit, and with no assumption about time scales. Physically, Eq. (67) treats each instant of emission as a flash sent from the position the source had at that instant. The lens turns each flash into its response ρ ^ / 2 π \hat{\rho}/2\pi ρ ^ /2 π Δ R ( t ) \Delta_R(t) Δ R ( t ) y ( t ) y(t) y ( t ) T T T (67) is thus a convolution whose kernel changes with the emission time, which is why no single amplification factor describes a moving source.
To make this more explicit, we can look at two limits of the system. With no lens, ρ ^ / 2 π \hat{\rho}/2\pi ρ ^ /2 π (67) reduces to the unlensed signal
h 0 ( T ) = q ( t 0 ) d t 0 d T , T = t 0 + Δ R ( t 0 ) (68) h_0(T)=q(t_0)\frac{\mathrm{d}t_0}{\mathrm{d}T},\qquad T=t_0+\Delta_R(t_0) \tag*{(68)} h 0 ( T ) = q ( t 0 ) d T d t 0 , T = t 0 + Δ R ( t 0 ) (68) which contains the Rømer delay of the emission time t 0 t_0 t 0 d t 0 / d T = ( 1 − n ^ ⋅ v s ) − 1 \mathrm{d}t_0/\mathrm{d}T=(1-\hat{\boldsymbol{n}}\cdot\boldsymbol{v}_{s})^{-1} d t 0 / d T = ( 1 − n ^ ⋅ v s ) − 1 v s \boldsymbol{v}_{s} v s ρ ^ \hat{\rho} ρ ^ ρ ^ ( τ ^ ; y ) = ∫ d 2 x δ ( τ ^ − T ( x , y ) ) \hat{\rho}(\hat{\tau};y)=\int\mathrm{d}^{2}x\,\delta(\hat{\tau}-\mathcal{T}(\boldsymbol{x},y)) ρ ^ ( τ ^ ; y ) = ∫ d 2 x δ ( τ ^ − T ( x , y )) x \boldsymbol{x} x t t t
T x ( t ) = t + Δ R ( t ) + 4 M T ( x , y ( t ) ) (69) T_{\boldsymbol{x}}(t)=t+\Delta_R(t)+4M\mathcal{T}(\boldsymbol{x},y(t)) \tag*{(69)} T x ( t ) = t + Δ R ( t ) + 4 M T ( x , y ( t )) (69) and the inverse function t x ( T ) t_{\boldsymbol{x}}(T) t x ( T ) t t t (67) then collapses onto it,
h ( T ) = 2 M π d d T ∫ d 2 x q ( t x ( T ) ) d t x d T (70) h(T)=\frac{2M}{\pi}\frac{\mathrm{d}}{\mathrm{d}T}\int\mathrm{d}^{2}x\,q(t_{\boldsymbol{x}}(T))\frac{\mathrm{d}t_{\boldsymbol{x}}}{\mathrm{d}T} \tag*{(70)} h ( T ) = π 2 M d T d ∫ d 2 x q ( t x ( T )) d T d t x (70) Each worldline therefore delivers the signal emitted at its own retarded time, with its own Doppler factor d t x / d T \mathrm{d}t_x/\mathrm{d}T d t x / d T 4 M T 4MT 4 M T − 4 M ln ∣ x ∣ -4M\ln|x| − 4 M ln ∣ x ∣ 4 M 4M 4 M q = e − i ω ˉ t q=e^{-i\bar{\omega}t} q = e − i ω ˉ t (70) gives back h = q ( T − Δ R ) F ( w , y ) h=q(T-\Delta_R)F(w,y) h = q ( T − Δ R ) F ( w , y ) Figure 8 illustrates the difference between the direct wave and the echo.
Figure 8. Each route has its own retarded time (schematic: edge-on orbit, speed exaggerated to v = 0.3 v=0.3 v = 0.3 T T T t 1 t_1 t 1 t 0 = t 1 + η t_0=t_1+\eta t 0 = t 1 + η Δ R = R sin i cos Φ \Delta_R=R\sin i\cos\Phi Δ R = R sin i cos Φ R R R − 1 -1 − 1
Comparison with the existing results
The exact formula (70) makes it possible to derive each approximate model and to see what it neglects. Most treatments of an orbiting source evaluate the static amplification factor at the instantaneous position of the source [36 , 108 ] . This model follows from Eq. (70) for a monochromatic source, q = e − i ω ˉ t q=e^{-i\bar{\omega}t} q = e − i ω ˉ t y ( t ) → y ( t 0 ) \mathbf{y}(t)\to\mathbf{y}(t_0) y ( t ) → y ( t 0 ) t 0 t_0 t 0 t x ≃ t 0 − 4 M T d t 0 / d T t_x\simeq t_0-4MT\,\mathrm{d}t_0/\mathrm{d}T t x ≃ t 0 − 4 M T d t 0 / d T
L a d ( T ) ≡ h ( T ) h 0 ( T ) ≃ F ( w l o c , y ( t 0 ) ) , w l o c = 4 M ω ˉ 1 − n ^ ⋅ v s ( t 0 ) , (71) \mathcal{L}_{\mathrm{ad}}(T)\equiv\frac{h(T)}{h_0(T)}\simeq F(w_{\mathrm{loc}},\mathbf{y}(t_0)),\qquad w_{\mathrm{loc}}=\frac{4M\bar{\omega}}{1-\hat{\mathbf{n}}\cdot\mathbf{v}_s(t_0)}, \tag*{(71)} L ad ( T ) ≡ h 0 ( T ) h ( T ) ≃ F ( w loc , y ( t 0 )) , w loc = 1 − n ^ ⋅ v s ( t 0 ) 4 M ω ˉ , (71) the static amplification factor at the retarded position of the source and at the Doppler-shifted frequency received. It amounts to assuming that the source configuration is invariant under time translations during the lensing delays. We know though that this assumption does not alway hold. Indeed, this approach misses a transit effect near conjunction. The worldlines in the diffraction spot have delays that differ by 4 M Δ τ ^ ≃ 8 M y 4M\Delta\hat{\tau} \simeq8My 4 M Δ τ ^ ≃ 8 M y d y / d t ≃ Ω ( R / 4 M ) 1 / 2 = 1 / ( 2 R ) dy/dt \simeq\Omega(R/4M)^{1/2} = 1/(2R) d y / d t ≃ Ω ( R /4 M ) 1/2 = 1/ ( 2 R ) y y y δ y ≃ 4 M y / R \delta y \simeq4My/R δ y ≃ 4 M y / R δ ln F ∼ w δ y \delta\ln F \sim w\,\delta y δ ln F ∼ w δ y w ≳ 1 w \gtrsim1 w ≳ 1 ≈ ( 1 – 2 ) w M / R \approx(1\text{--}2)\,wM/R ≈ ( 1 – 2 ) w M / R
Δ R + η = R sin i cos Φ + R ( 1 − sin i cos Φ ) = R (72) \Delta_R+\eta=R\sin i\cos\Phi+R(1-\sin i\cos\Phi)=R \tag*{(72)} Δ R + η = R sin i cos Φ + R ( 1 − sin i cos Φ ) = R (72) exactly, so the echo arrives at a fixed delay R R R Figure 8 ). The adiabatic model imposes the Doppler shift of the direct wave on the echo, and its echo phase is in error by ( ω l o c − ω ˉ ) η (\omega_{\rm loc}-\bar{\omega})\eta ( ω loc − ω ˉ ) η ≃ 0.33 w ( R / M ) 1 / 2 r a d \simeq0.33w(R/M)^{1/2}\,\mathrm{rad} ≃ 0.33 w ( R / M ) 1/2 rad Φ ≃ 2 π / 3 \Phi\simeq2\pi/3 Φ ≃ 2 π /3 3.3 w r a d 3.3w\,\mathrm{rad} 3.3 w rad R = 100 M R=100M R = 100 M 10 w r a d 10w\,\mathrm{rad} 10 w rad R = 10 3 M R=10^3M R = 1 0 3 M w ( R / M ) 1 / 2 ≳ 3 w(R/M)^{1/2}\gtrsim3 w ( R / M ) 1/2 ≳ 3
A periodic source and the lensing modulation
The time-domain formula (67) holds for any source. For the simplest one, a monochromatic source q = q 0 e − i ω ˉ t q=q_0e^{-i\bar{\omega}t} q = q 0 e − i ω ˉ t (60) into Eq. (65) and use the arrival time T T T
h ( T ) = q 0 ∫ d ω 2 π e − i ω T ∫ d t e i ( ω − ω ˉ ) t g ω ( t ) , g ω ( t ) = e − i ω n ^ ⋅ x s ( t ) F ( 4 M ω , y ( t ) ) (73) h(T)=q_0\int\frac{d\omega}{2\pi}e^{-i\omega T}\int dt\,e^{i(\omega-\bar{\omega})t}g_\omega(t),\qquad g_\omega(t)=e^{-i\omega\hat{\boldsymbol{n}}\cdot\boldsymbol{x}_s(t)}F(4M\omega,y(t)) \tag*{(73)} h ( T ) = q 0 ∫ 2 π d ω e − iω T ∫ d t e i ( ω − ω ˉ ) t g ω ( t ) , g ω ( t ) = e − iω n ^ ⋅ x s ( t ) F ( 4 M ω , y ( t )) (73) where g ω ( t ) g_\omega(t) g ω ( t ) t t t g ω g_\omega g ω P = 2 π / Ω P=2\pi/\Omega P = 2 π /Ω t t t ω = ω ˉ + n Ω \omega=\bar{\omega}+n\Omega ω = ω ˉ + n Ω
h ( T ) = q 0 ∑ n c n e − i ω n T , c n = 1 P ∫ 0 P d t e i n Ω t e − i ω n n ^ ⋅ x s ( t ) F ( 4 M ω n , y ( t ) ) (74) h(T)=q_0\sum_n c_n e^{-i\omega_nT},\qquad c_n=\frac{1}{P}\int_0^Pdt\,e^{in\Omega t}e^{-i\omega_n\hat{\boldsymbol{n}}\cdot\boldsymbol{x}_s(t)}F(4M\omega_n,y(t)) \tag*{(74)} h ( T ) = q 0 n ∑ c n e − i ω n T , c n = P 1 ∫ 0 P d t e in Ω t e − i ω n n ^ ⋅ x s ( t ) F ( 4 M ω n , y ( t )) (74) with ω n = ω ˉ + n Ω \omega_n=\bar{\omega}+n\Omega ω n = ω ˉ + n Ω F = 1 F=1 F = 1 c n ( 0 ) = i n J n ( ω n R sin i ) c_n^{(0)}=i^nJ_n(\omega_nR\sin i) c n ( 0 ) = i n J n ( ω n R sin i ) ω ˉ \bar{\omega} ω ˉ ∣ n ∣ ≲ ω ˉ R sin i |n|\lesssim\bar{\omega}R\sin i ∣ n ∣ ≲ ω ˉ R sin i Figure 9 (b)]. Since the comb and Eq. (67) are the same convolution evaluated in frequency and in time, their comparison tests the whole construction: they agree to 3.2 × 10 − 5 3.2\times10^{-5} 3.2 × 1 0 − 5 3 × 10 − 5 r a d 3\times10^{-5}\,\mathrm{rad} 3 × 1 0 − 5 rad
Figure 9. Exact bound-lens result for w = 1 w = 1 w = 1 R = 100 M R = 100M R = 100 M cos i = 0.1 \cos i = 0.1 cos i = 0.1 F ( w , y ) F(w,y) F ( w , y ) 1 ± 2 M / η 1 \pm2M/\eta 1 ± 2 M / η (74) , lensed and unlensed. (c) Phase and (d) log-amplitude of the lensing modulation (75) , with the orbit average removed, against the arrival phase; the insets zoom on conjunction. Solid: exact; dashed: adiabatic model (71) ; dotted: eikonal model.
The observable is the lensing modulation
L ( T ) = h ( T ) h 0 ( T ) (75) \mathcal{L}(T)=\frac{h(T)}{h_0(T)} \tag*{(75)} L ( T ) = h 0 ( T ) h ( T ) (75) the ratio of the lensed signal to the unlensed one (68) . Figure 9 shows it for w = 1 w=1 w = 1 R = 100 M R=100M R = 100 M cos i = 0.1 \cos i=0.1 cos i = 0.1 L \mathcal{L} L ∣ n ∣ ≤ 2 |n|\leq2 ∣ n ∣ ≤ 2 Q ϕ Q_{\phi} Q ϕ Q ln A Q_{\ln A} Q l n A L / L e i k \mathcal{L}/\mathcal{L}_{\mathrm{eik}} L / L eik Q ϕ b e y o n d S h Q_{\phi}^{\mathrm{beyond\ Sh}} Q ϕ beyond Sh
Results
With the exact signal at hand we can now measure how well each approximate description reproduces it, and where along the orbit each of them breaks down. Besides the adiabatic model (71) we consider the eikonal model, the paraxial model and two forms of geometric optics. Geometric optics does not require the adiabatic step. The delay density (17) is singular at the arrival of each image, with a step at the minimum and a logarithm at the saddle, and keeping only these singularities in Eq. (67) gives the time-domain form of geometric optics,
h G O ( T ) = μ + ( t + ) q ( t + ) d t + d T − μ − ( t − ) H [ q ( t − ) d t − d T ] , (76) h_{\mathrm{GO}}(T)=\sqrt{\mu_{+}(t_{+})q(t_{+})}\frac{\mathrm{d}t_{+}}{\mathrm{d}T}-\sqrt{\mu_{-}(t_{-})}\mathcal{H}\left[q(t_{-})\frac{\mathrm{d}t_{-}}{\mathrm{d}T}\right], \tag*{(76)} h GO ( T ) = μ + ( t + ) q ( t + ) d T d t + − μ − ( t − ) H [ q ( t − ) d T d t − ] , (76) where the retarded times t ± t_{\pm} t ± T = t ± + Δ R ( t ± ) + 4 M T ± ( y ( t ± ) ) T=t_{\pm}+\Delta_{R}(t_{\pm})+4M\mathcal{T}_{\pm}(y(t_{\pm})) T = t ± + Δ R ( t ± ) + 4 M T ± ( y ( t ± )) T ± \mathcal{T}_{\pm} T ± H \mathcal{H} H − i -\mathrm{i} − i 4 M T − ≃ η + 2 M ln ( η / 2 M ) + 4 M 4M\mathcal{T}_{-}\simeq\eta+2M\ln(\eta/2M)+4M 4 M T − ≃ η + 2 M ln ( η /2 M ) + 4 M η p a r \eta_{\mathrm{par}} η par η \eta η L e i k = y − i w l o c \mathcal{L}_{\mathrm{eik}}=y^{-\mathrm{i}w_{\mathrm{loc}}} L eik = y − i w loc Q ϕ Q_{\phi} Q ϕ § 7.1 .
The exact modulation
The exact modulation for w = 1 w=1 w = 1 R = 100 M R=100M R = 100 M cos i = 0.1 \cos i=0.1 cos i = 0.1 Figure 9 . It has two distinct components: the flare at conjunction, where arg L \arg\mathcal{L} arg L ln ∣ L ∣ \ln|\mathcal{L}| ln ∣ L ∣ Table 2 lists the non-degenerate modulation and the errors of the models for a subset of the configurations. The phase modulation grows roughly linearly with w w w Q ϕ = 0.03 Q_{\phi}=0.03 Q ϕ = 0.03 0.08 0.08 0.08 w = 0.3 w=0.3 w = 0.3 w = 10 w=10 w = 10 Q ϕ Q_{\phi} Q ϕ w ≥ 1 w\ge1 w ≥ 1 w w w
Table 2. Bound lens over one orbit. Columns 4–6: rms of the non-degenerate phase (rad) and log-amplitude of the exact modulation, and the phase content beyond the Shapiro delay (equal to the error of the eikonal model). Columns 7–10: rms non-degenerate phase error (rad) of the approximate models.
Near conjunction: diffraction
Near conjunction the lensing signal beyond the Shapiro delay is 0.044–0.21 rad at w = 0.3 w=0.3 w = 0.3 w = 1 w=1 w = 1 w = 0.3 w=0.3 w = 0.3 Figure 10 ). At w = 1 w = 1 w = 1 ∣ F G O / F − 1 ∣ = 0.15 |F_{\mathrm{GO}}/F - 1| = 0.15 ∣ F GO / F − 1∣ = 0.15 w = 1 w = 1 w = 1 y ≃ 0.5 y \simeq0.5 y ≃ 0.5 β G O \beta_{\mathrm{GO}} β GO (63) , the wave does not separate into two classical worldlines, and no choice of retarded times can repair this.
Figure 10. Near conjunction: the exact modulation (black) and the five models, for R = 100 M R = 100M R = 100 M cos i = 0.1 \cos i = 0.1 cos i = 0.1 w = 0.3 w = 0.3 w = 0.3 arg L \arg\mathcal{L} arg L ln ∣ L ∣ \ln|\mathcal{L}| ln ∣ L ∣ ∣ n ∣ ≤ 2 |n| \le2 ∣ n ∣ ≤ 2 w = 0.3 w = 0.3 w = 0.3
The adiabatic model keeps the exact static amplification factor, and with it all the diffraction, and is the most accurate approximation near conjunction for w ≲ 1 w \lesssim1 w ≲ 1 w = 1 w = 1 w = 1 × 10 − 2 \times10^{-2} × 1 0 − 2 R = 100 M R = 100M R = 100 M × 10 − 3 \times10^{-3} × 1 0 − 3 R = 10 3 M R = 10^{3}M R = 1 0 3 M § 7 : the worldlines of the diffraction spot have different delays and see the source at slightly different offsets, an error of order w M / R wM/R w M / R
The far side: geometry and the echo
On the far side diffraction is no longer the issue, and the models fail through the geometry and the timing. The paraxial model evaluates the Shapiro phase with the wrong geometry, ν ln ( η / η p a r ) \nu\ln(\eta/\eta_{\mathrm{par}}) ν ln ( η / η par ) w w w w = 0.3 w = 0.3 w = 0.3 w = 10 w = 10 w = 10 w ≥ 1 w \ge1 w ≥ 1 ( 1.5 (1.5 ( 1.5 1.7 ) × 10 − 2 1.7) \times10^{-2} 1.7 ) × 1 0 − 2 R = 100 M R = 100M R = 100 M ( 1.2 (1.2 ( 1.2 1.6 ) × 10 − 3 1.6) \times10^{-3} 1.6 ) × 1 0 − 3 R = 10 3 M R = 10^{3}M R = 1 0 3 M w w w § 7 ).
Geometric optics at the images’ own retarded times, on the contrary, delivers the echo at the right time and with the right Doppler factor, and on the far side the echo is a well separated classical worldline. For w ≥ 1 w \ge1 w ≥ 1 3.8 × 10 − 5 3.8 \times10^{-5} 3.8 × 1 0 − 5 3.3 × 10 − 3 3.3 \times10^{-3} 3.3 × 1 0 − 3 w ≥ 3 w \ge3 w ≥ 3 3.4 × 10 − 4 3.4 \times10^{-4} 3.4 × 1 0 − 4 8.9 × 10 − 3 8.9 \times10^{-3} 8.9 × 1 0 − 3 Figure 11 ). For w ≳ 3 w \gtrsim3 w ≳ 3 w ≲ 1 w \lesssim1 w ≲ 1
Figure 11. Error budget against w w w ∣ g ∣ = 1 |g| = 1 ∣ g ∣ = 1 y < 3 y < 3 y < 3 β > π / 2 \beta> \pi/2 β > π /2 R = 100 M R = 100M R = 100 M cos i = 0.1 \cos i = 0.1 cos i = 0.1 R = 10 3 M R = 10^3M R = 1 0 3 M cos i = 0.02 \cos i = 0.02 cos i = 0.02
Beyond the scalar linearized model
All the results above assume the model of § 6.1 : a scalar wave in the linearized field of a static lens, emitted isotropically. Relaxing these assumptions does not change the structure of the calculation. The corrections enter only through the static inputs of the time-domain formula, the amplification factor (or the delay density ρ ^ \hat{\rho} ρ ^ q q q (67) , with its retarded times and Doppler factors, is unchanged. It is therefore enough to estimate how much each correction changes the static inputs, and to compare it with the lensing signal and with the errors of the models.
The largest correction comes from the field of the lens itself. On the exact Schwarzschild metric the scalar wave of a point source at finite distance separates into partial waves that obey the Regge–Wheeler equation with s = 0 s = 0 s = 0 [86 ] , and the sum converges absolutely, because for ℓ > ω R \ell> \omega R ℓ > ω R Θ = 4 M / b + 15 π M 2 / 4 b 2 \Theta= 4M/b + 15\pi M^{2}/4b^{2} Θ = 4 M / b + 15 π M 2 /4 b 2 ϵ / x \epsilon/x ϵ / x ϵ = 15 π 32 ( M / R ) 1 / 2 \epsilon= \frac{15\pi}{32}(M/R)^{1/2} ϵ = 32 15 π ( M / R ) 1/2 R = 100 M R = 100M R = 100 M R = 10 3 M R = 10^{3}M R = 1 0 3 M R = 100 M R = 100M R = 100 M b ≤ 2 M b \le2M b ≤ 2 M b → 3 3 M + b \to3\sqrt{3}M^{+} b → 3 3 M + § 6.2 . A rotating (Kerr) lens adds a term ≃ 0.34 a / M \simeq0.34a/M ≃ 0.34 a / M [89 ] .
The spin of the wave matters less. In the long-wavelength regime the helicity-preserving and helicity-reversing scattering amplitudes of GWs on Schwarzschild are f s cos 4 ( θ / 2 ) f_{s}\cos^{4}(\theta/2) f s cos 4 ( θ /2 ) f s sin 4 ( θ / 2 ) f_{s}\sin^{4}(\theta/2) f s sin 4 ( θ /2 ) f s f_{s} f s [33 , 35 , 80 , 107 ] . For the bound lens the scattering angle is β \beta β ∣ δ F + + ∣ = ( 1 + cos 2 β 2 ) M / R |\delta F_{++}| = (1 + \cos^{2}\frac{\beta}{2})M/R ∣ δ F ++ ∣ = ( 1 + cos 2 2 β ) M / R ≃ M / R \simeq M/R ≃ M / R w w w
The emission pattern of the source enters in a different way. Each worldline leaves the source in its own direction, so the two images see the binary from slightly different directions.
With g g g ∣ g ∣ = 2 2 tan ( ι / 2 ) |g| = 2\sqrt{2}\tan(\iota/2) ∣ g ∣ = 2 2 tan ( ι /2 ) ι \iota ι ( m / M ) v (m/M)v ( m / M ) v
For w ≳ 1 w \gtrsim1 w ≳ 1 R = 10 2 R = 10^2 R = 1 0 2 10 3 M 10^3M 1 0 3 M
w ( M R ) 1 / 2 , ∣ g ∣ w ( M R ) 1 / 2 > w M R ∼ M R (77) w\left(\frac{M}{R}\right)^{1/2},\quad|g|w\left(\frac{M}{R}\right)^{1/2} > w\frac{M}{R} \sim\frac{M}{R} \tag*{(77)} w ( R M ) 1/2 , ∣ g ∣ w ( R M ) 1/2 > w R M ∼ R M (77) and Table 3 collects the budget at w = 1 w = 1 w = 1 0.04 0.04 0.04 0.4 r a d 0.4\,\mathrm{rad} 0.4 rad w ≤ 1 w \le1 w ≤ 1 ≃ 1.5 \simeq1.5 ≃ 1.5 ∣ g ∣ ≲ 1 |g| \lesssim1 ∣ g ∣ ≲ 1 w ≃ 0.3 w \simeq0.3 w ≃ 0.3
Table 3. Error budget at w = 1 w = 1 w = 1 R = 100 M R = 100M R = 100 M R = 10 3 M R = 10^{3}M R = 1 0 3 M
Chirping sources, distinguishability and the astrophysical window
The time-domain formula (67) holds for any q ( t ) q(t) q ( t ) 10 + 10 M ⊙ 10 + 10\,M_{\odot} 10 + 10 M ⊙ 800 M ⊙ 800M_{\odot} 800 M ⊙ w = 1 w = 1 w = 1 f ≃ 10 H z f \simeq10\,\mathrm{Hz} f ≃ 10 Hz 0.8 ≤ w ≤ 3.5 0.8 \leq w \leq3.5 0.8 ≤ w ≤ 3.5 f ≃ 8 f \simeq8 f ≃ 8 35 H z 35\,\mathrm{Hz} 35 Hz R = 100 M R = 100M R = 100 M Figure 12 ). At the conjunction with w l o c ≃ 1 w_{\mathrm{loc}} \simeq1 w loc ≃ 1 y min = 0.5 y_{\min} = 0.5 y m i n = 0.5 3 × 10 − 4 3 \times10^{-4} 3 × 1 0 − 4 M M M ( 3 − 5 ) × 10 − 2 (3\!-\!5) \times10^{-2} ( 3 − 5 ) × 1 0 − 2
Figure 12. Chirping source: a 10 + 10 M ⊙ 10+10M_{\odot} 10 + 10 M ⊙ 800 M ⊙ 800M_{\odot} 800 M ⊙ R = 100 M R=100M R = 100 M cos i = 0.1 \cos i=0.1 cos i = 0.1 0.8 ≤ w ≤ 3.5 0.8\le w\le3.5 0.8 ≤ w ≤ 3.5 f ≃ 8 f\simeq8 f ≃ 8 35 H z 35\ \mathrm{Hz} 35 Hz ∣ h ∣ → 0 |h|\to0 ∣ h ∣ → 0 ∣ h ∣ |h| ∣ h ∣ w l o c w_{\mathrm{loc}} w loc w l o c ≃ 1 w_{\mathrm{loc}}\simeq1 w loc ≃ 1 1.5 1.5 1.5 h m o d e l / h h_{\mathrm{model}}/h h model / h 1 s ≃ 250 M 1\ \mathrm{s}\simeq250M 1 s ≃ 250 M
Whether these differences can be measured depends on the signal-to-noise ratio. Two signals are distinguishable when their optimally aligned difference has ⟨ δ h ∣ δ h ⟩ > 1 \langle\delta h|\delta h\rangle> 1 ⟨ δ h ∣ δ h ⟩ > 1 [65 ] , i.e. above the signal-to-noise ratio ρ c r i t = ( 2 M ⊥ ) − 1 / 2 \rho_{\mathrm{crit}} = (2\mathcal{M}_{\perp})^{-1/2} ρ crit = ( 2 M ⊥ ) − 1/2 M ⊥ \mathcal{M}_{\perp} M ⊥ w = 1 w = 1 w = 1 3 3 3 y min ≲ 1.5 y_{\min} \lesssim1.5 y m i n ≲ 1.5 ρ ≃ 2 \rho\simeq2 ρ ≃ 2 7 7 7 ρ ≃ 4 \rho\simeq4 ρ ≃ 4 16 16 16 R = 100 M R = 100M R = 100 M w ≥ 1 w \geq1 w ≥ 1 ρ ≃ 10 \rho\simeq10 ρ ≃ 10 35 35 35 R = 10 3 M R = 10^{3}M R = 1 0 3 M ρ ≳ 110 \rho\gtrsim110 ρ ≳ 110 ρ ≃ 15 \rho\simeq15 ρ ≃ 15 30 30 30 w ≲ 1.5 w \lesssim1.5 w ≲ 1.5 Table 4 , Figure 13 ). The lens is therefore detectable at modest signal-to-noise ratios, and its diffraction beyond the Shapiro delay at only slightly larger ones.
Table 4. Critical signal-to-noise ratio ρ c r i t \rho_{\mathrm{crit}} ρ crit ≲ 3 \lesssim3 ≲ 3
Figure 13. Critical signal-to-noise ratio ρ c r i t \rho_{\mathrm{crit}} ρ crit y min = [ R ( 1 − sin i ) / 2 M ] 1 / 2 ≃ ( R / 4 M ) 1 / 2 ∣ cos i ∣ y_{\min} = [R(1-\sin i)/2M]^{1/2} \simeq(R/4M)^{1/2}|\cos i| y m i n = [ R ( 1 − sin i ) /2 M ] 1/2 ≃ ( R /4 M ) 1/2 ∣ cos i ∣ y min = 0 y_{\min}=0 y m i n = 0 R = 100 M R=100M R = 100 M R = 10 3 M R=10^3M R = 1 0 3 M ρ > ρ c r i t = ( 2 M ⊥ ) − 1 / 2 \rho>\rho_{\mathrm{crit}}=(2\mathcal{M}_{\perp})^{-1/2} ρ > ρ crit = ( 2 M ⊥ ) − 1/2 M ⊥ \mathcal{M}_{\perp} M ⊥ ≲ 3 \lesssim3 ≲ 3 ρ = 10 \rho=10 ρ = 10 y min y_{\min} y m i n w w w R = 100 M R=100M R = 100 M y min ≲ 1.5 y_{\min}\lesssim1.5 y m i n ≲ 1.5 w = 0.3 w=0.3 w = 0.3 − w ln y -w\ln y − w ln y R = 10 3 M R=10^3M R = 1 0 3 M ρ ≲ 10 \rho\lesssim10 ρ ≲ 10 y min ≃ 5 y_{\min}\simeq5 y m i n ≃ 5 w = 1 w=1 w = 1 w = 3 w=3 w = 3 y min ≲ 1 y_{\min}\lesssim1 y m i n ≲ 1
It remains to see where such systems can be found. Since w ≃ 1.24 × 10 − 4 ( M z / M ⊙ ) ( f / H z ) w \simeq1.24 \times10^{-4}(M_{z}/M_{\odot})(f/\mathrm{Hz}) w ≃ 1.24 × 1 0 − 4 ( M z / M ⊙ ) ( f / Hz ) M z M_{z} M z 0.3 ≤ w ≤ 3 0.3 \leq w \leq3 0.3 ≤ w ≤ 3 M ≃ ( 0.24 M \simeq(0.24 M ≃ ( 0.24 2.4 ) × 10 4 M ⊙ ( f / H z ) − 1 2.4) \times10^{4}M_{\odot}(f/\mathrm{Hz})^{-1} 2.4 ) × 1 0 4 M ⊙ ( f / Hz ) − 1 Figure 14 ), and Table 5 gives the relevant time scales at w = 1 w = 1 w = 1 R = 100 R = 100 R = 100 10 3 M 10^{3}M 1 0 3 M 30 + 30 M ⊙ 30 + 30\,M_{\odot} 30 + 30 M ⊙ ∼ 800 M ⊙ \sim800M_{\odot} ∼ 800 M ⊙ y c y_{c} y c P = 2 y c ( M / R ) 1 / 2 P = 2y_{c}(M/R)^{1/2} P = 2 y c ( M / R ) 1/2 w = 1 w = 1 w = 1 ρ = 10 \rho= 10 ρ = 10 30 30 30 P ≃ 0.2 P \simeq0.2 P ≃ 0.2 0.6 0.6 0.6 R = 100 M R = 100M R = 100 M 0.04 0.04 0.04 0.14 0.14 0.14 R = 10 3 M R=10^{3}M R = 1 0 3 M ∼ 10 3 M \sim10^{3}M ∼ 1 0 3 M [11 , 53 , 90 ] .
Table 5. Diffraction-regime examples at w = 1 w = 1 w = 1 P P P 2 R 2R 2 R
Figure 14. The diffraction regime 0.3 ≤ w ≤ 3 0.3 \le w \le3 0.3 ≤ w ≤ 3
Conclusions
The main results of this paper are an exact solution of the point-mass lens beyond the diffraction integral, obtained with the proper-time representation of the propagator, and its application to the bound lens, a configuration in which the approximations behind the diffraction integral fail together and which therefore serves as a benchmark for the approximations used in the literature.
For a point mass, the Duru–Kleinert clock, which measures the Shapiro delay accumulated along each worldline, and the Kustaanheimo–Stiefel map make the proper-time path integral Gaussian and give the propagator in closed form in the linearized field, at every frequency, for every position of source and observer and for every scattering angle. Nothing is fixed by hand in this construction: the DK time and the fibre angle are integrated over, and stationarity with respect to them reproduces the Hamiltonian and the fibre constraints. For an observer at infinity these two variables span a plane, with the DK time as radius, the fibre angle as azimuth and the Fermat potential as phase, and the exact far-field amplitude takes the form of the point-lens amplification factor F ( w , y ) F(w,y) F ( w , y ) y 2 = η / 2 M y^{2} = \eta/2M y 2 = η /2 M η \eta η w ≫ 1 w \gg1 w ≫ 1 w y ≫ 1 wy \gg1 w y ≫ 1 ε E ≃ λ ˉ / d + G M / c 2 d + O ( sin 2 θ / 2 ) \varepsilon_E \simeq\bar{\lambda}/d + GM/c^{2}d + O(\sin^{2}\theta/2) ε E ≃ λ ˉ / d + GM / c 2 d + O ( sin 2 θ /2 )
In the bound lens all these approximations fail at once in the diffraction regime, and we have computed its signal exactly for a scalar wave in the linearized field of a static point lens. For a source held fixed, the exact far-field amplitude keeps the point-lens form at every orbital phase, with the offset given by the exact excess path, whereas the paraxial geometry, which replaces η \eta η η \eta η π 2 / 4 \pi^{2}/4 π 2 /4
The comparison with the exact signal sorts these approximations by the physics they miss. Most of the phase modulation is the Shapiro delay, which an orbital timing model absorbs, and the lensing beyond it, 8–60% of the phase modulation for w ≥ 1 w \ge1 w ≥ 1 w = 0.3 w = 0.3 w = 0.3 w = 1 w = 1 w = 1 ≈ ( 1 – 2 ) w M / R \approx(1\text{--}2)wM/R ≈ ( 1 – 2 ) w M / R w w w w ≥ 1 w \ge1 w ≥ 1 10 − 2 10^{-2} 1 0 − 2 w ≥ 3 w \ge3 w ≥ 3 w ∼ 1 w \sim1 w ∼ 1 ρ ≃ 4 – 16 \rho\simeq4\text{--}16 ρ ≃ 4 – 16 w = 1 – 3 w = 1\text{--}3 w = 1 – 3 10 2 – 10 3 M 10^{2}\text{--}10^{3}M 1 0 2 – 1 0 3 M
Several extensions follow naturally. The partial-wave construction carries over to spin 2 and to Kerr lenses, and a worldline derivation with spin would use the N = 4 \mathcal{N} = 4 N = 4 [10 , 70 ] ; whether a gauge of Duru–Kleinert type makes it solvable is open. For extended lenses no Gaussian gauge is known, but the proper-time representation still provides an exact starting point from which the approximations can be taken one at a time and their validity judged. The point mass and the bound lens show how much is lost when they are taken together.
Acknowledgments
This work was supported by a grant from the Simons Foundation (00017375, RJ). This work was in part supported by the European Research Council (ERC Synergy Grant RedHOT, Grant Agreement No. 101225120). Funded by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
AI usage: Claude Code was used to assist with coding, computations, and graphic generation. The ideas, inferences, and results all originate in the (human) brains of the authors. Claude and ChatGPT were used to check citations, spelling, grammar, and inconsistencies. The text was written by the authors, who are responsible for the results, findings, and conclusions of this article.
The proper-time and Duru–Kleinert calculations
With p ^ = − ( i / ω ) ∇ \hat{p}=-(i/\omega)\nabla p ^ = − ( i / ω ) ∇ ⟨ x ∣ e − i ω τ p ^ 2 ∣ x ′ ⟩ = ( ω / 4 π i τ ) 3 / 2 e i ω d 2 / 4 τ \langle x|e^{-i\omega\tau\hat{p}^{2}}|x'\rangle=(\omega/4\pi i\tau)^{3/2}e^{i\omega d^{2}/4\tau} ⟨ x ∣ e − iω τ p ^ 2 ∣ x ′ ⟩ = ( ω /4 π i τ ) 3/2 e iω d 2 /4 τ d = ∣ x − x ′ ∣ d=|x-x'| d = ∣ x − x ′ ∣ p = ϵ − i ω p=\epsilon-i\omega p = ϵ − iω q = ϵ ′ − i ω d 2 / 4 q=\epsilon'-i\omega d^{2}/4 q = ϵ ′ − iω d 2 /4 ϵ , ϵ ′ → 0 + \epsilon,\epsilon'\to0^{+} ϵ , ϵ ′ → 0 +
∫ 0 ∞ d τ τ − 3 / 2 e − p τ − q / τ = π q e − 2 p q , Re p , Re q > 0. (78) \int_{0}^{\infty}d\tau\,\tau^{-3/2}e^{-p\tau-q/\tau}=\sqrt{\frac{\pi}{q}}e^{-2\sqrt{pq}},\qquad\operatorname{Re}p,\operatorname{Re}q>0. \tag*{(78)} ∫ 0 ∞ d τ τ − 3/2 e − p τ − q / τ = q π e − 2 pq , Re p , Re q > 0. (78) As the regulators vanish, p q → − ω 2 d 2 / 4 pq\to-\omega^{2}d^{2}/4 pq → − ω 2 d 2 /4 p q → − i ω d / 2 \sqrt{pq}\to-i\omega d/2 pq → − iω d /2 e − 2 p q → e i ω d e^{-2\sqrt{pq}}\to e^{i\omega d} e − 2 pq → e iω d π / q = ( 2 π / d ω ) e i π / 4 \sqrt{\pi/q}=(2\sqrt{\pi}/d\sqrt{\omega})e^{i\pi/4} π / q = ( 2 π / d ω ) e iπ /4 ( ω / 4 π i ) 3 / 2 = ω 3 / 2 ( 4 π ) − 3 / 2 e − 3 i π / 4 (\omega/4\pi i)^{3/2}=\omega^{3/2}(4\pi)^{-3/2}e^{-3i\pi/4} ( ω /4 π i ) 3/2 = ω 3/2 ( 4 π ) − 3/2 e − 3 iπ /4
G 0 = i ω ω 3 / 2 ( 4 π ) − 3 / 2 e − 3 i π / 4 2 π d ω e i π / 4 e i ω d = e i ω d 4 π d , (79) G_{0}=\frac{i}{\omega}\,\omega^{3/2}(4\pi)^{-3/2}e^{-3i\pi/4}\frac{2\sqrt{\pi}}{d\sqrt{\omega}}e^{i\pi/4}e^{i\omega d}=\frac{e^{i\omega d}}{4\pi d}, \tag*{(79)} G 0 = ω i ω 3/2 ( 4 π ) − 3/2 e − 3 iπ /4 d ω 2 π e iπ /4 e iω d = 4 π d e iω d , (79) which we also checked by direct quadrature at finite regulator. The exponent ω ( τ + d 2 / 4 τ ) \omega(\tau+d^{2}/4\tau) ω ( τ + d 2 /4 τ ) τ ∗ = d / 2 \tau_{*}=d/2 τ ∗ = d /2 4 ω / d 4\omega/d 4 ω / d ( 2 π / ω d ) 1 / 2 (2\pi/\omega d)^{1/2} ( 2 π / ω d ) 1/2 § 3 , while the position of the saddle does not depend on the frequency.
The same projector z z † = 1 2 ( r + x ⋅ σ ) zz^{\dagger}=\frac{1}{2}(r+x\cdot\sigma) z z † = 2 1 ( r + x ⋅ σ )
∣ z † z ′ ∣ 2 = z ′ † ( z z † ) z ′ = 1 2 ( r r ′ + x ⋅ x ′ ) ≡ ϖ 2 . (80) |z^{\dagger}z'|^{2}=z'^{\dagger}(zz^{\dagger})z'=\frac{1}{2}(rr'+x\cdot x')\equiv\varpi^{2}. \tag*{(80)} ∣ z † z ′ ∣ 2 = z ′ † ( z z † ) z ′ = 2 1 ( r r ′ + x ⋅ x ′ ) ≡ ϖ 2 . (80) For the Laplacian we use ∇ u 2 = 4 ∑ a ∂ z a ∂ z ˉ a \nabla_{u}^{2}=4\sum_{a}\partial_{z_{a}}\partial_{\bar{z}_{a}} ∇ u 2 = 4 ∑ a ∂ z a ∂ z ˉ a ∂ z ˉ a x i = ( σ i z ) a \partial_{\bar{z}_{a}}x_{i}=(\sigma_{i}z)_{a} ∂ z ˉ a x i = ( σ i z ) a ∂ z a x i = ( z † σ i ) a \partial_{z_{a}}x_{i}=(z^{\dagger}\sigma_{i})_{a} ∂ z a x i = ( z † σ i ) a f ( x ( u ) ) f(x(u)) f ( x ( u )) ∑ a ∂ z a ∂ z ˉ a f = tr ( σ i ) ∂ i f + z † σ j σ i z ∂ i ∂ j f \sum_{a}\partial_{z_{a}}\partial_{\bar{z}_{a}}f=\operatorname{tr}(\sigma_{i})\partial_{i}f+z^{\dagger}\sigma_{j}\sigma_{i}z\,\partial_{i}\partial_{j}f ∑ a ∂ z a ∂ z ˉ a f = tr ( σ i ) ∂ i f + z † σ j σ i z ∂ i ∂ j f z † σ j σ i z = δ i j r + i ϵ j i k x k z^{\dagger}\sigma_{j}\sigma_{i}z=\delta_{ij}r+i\epsilon_{jik}x_{k} z † σ j σ i z = δ ij r + i ϵ j ik x k ∂ i ∂ j \partial_{i}\partial_{j} ∂ i ∂ j
∇ u 2 = 4 r ∇ x 2 , (81) \nabla_{u}^{2}=4r\,\nabla_{x}^{2}, \tag*{(81)} ∇ u 2 = 4 r ∇ x 2 , (81) which is the identity behind Eq. (34) . The volume elements are related by d 4 u = J d 3 x d θ d^{4}u=J\,d^{3}x\,d\theta d 4 u = J d 3 x d θ S U ( 2 ) SU(2) S U ( 2 ) u u u x x x J J J r r r u → λ u u\to\lambda u u → λ u d 4 u → λ 4 d 4 u d^{4}u\to\lambda^{4}d^{4}u d 4 u → λ 4 d 4 u d 3 x → λ 6 d 3 x d^{3}x\to\lambda^{6}d^{3}x d 3 x → λ 6 d 3 x r → λ 2 r r\to\lambda^{2}r r → λ 2 r J ( λ 2 r ) = λ − 2 J ( r ) J(\lambda^{2}r)=\lambda^{-2}J(r) J ( λ 2 r ) = λ − 2 J ( r ) J = C / r J=C/r J = C / r ∣ u ∣ 2 < L |u|^{2}<L ∣ u ∣ 2 < L π 2 L 2 / 2 \pi^{2}L^{2}/2 π 2 L 2 /2 ∫ r < L d 3 x ∫ 0 2 π d θ C / r = 4 π 2 C L 2 \int_{r<L}d^{3}x\int_{0}^{2\pi}d\theta\,C/r=4\pi^{2}CL^{2} ∫ r < L d 3 x ∫ 0 2 π d θ C / r = 4 π 2 C L 2 C = 1 / 8 C=1/8 C = 1/8
d 4 u = d 3 x d θ 8 r . (82) d^{4}u=\frac{d^{3}x\,d\theta}{8r}. \tag*{(82)} d 4 u = 8 r d 3 x d θ . (82) Because the KS map is not one to one, the oscillator Green function g ( u , u ′ ) g(u,u') g ( u , u ′ ) (36) , which solves ω 2 ( h ^ − 4 M ) g = δ 4 ( u − u ′ ) \omega^{2}(\hat{h}-4M)g=\delta^{4}(u-u') ω 2 ( h ^ − 4 M ) g = δ 4 ( u − u ′ ) u ′ u' u ′ x ′ x' x ′ φ 0 \varphi_{0} φ 0 e i θ ′ u ′ ) = ∫ 0 2 π d θ ′ φ 0 ( x ′ ) = 2 π φ 0 ( x ′ ) e^{i\theta'}u') = \int_{0}^{2\pi}d\theta'\,\varphi_0(x') = 2\pi\varphi_0(x') e i θ ′ u ′ ) = ∫ 0 2 π d θ ′ φ 0 ( x ′ ) = 2 π φ 0 ( x ′ ) 2 π 8 ∫ d 3 x φ 0 T / r \frac{2\pi}{8}\int d^3x\,\varphi_0\,T/r 8 2 π ∫ d 3 x φ 0 T / r T T T θ ′ \theta' θ ′ δ \delta δ
∫ 0 2 π d θ ′ δ 4 ( u − e i θ ′ u ′ ) = 8 r δ 3 ( x − x ′ ) (83) \int_{0}^{2\pi}d\theta'\,\delta^4(u-e^{i\theta'}u')=8r\,\delta^3(x-x') \tag*{(83)} ∫ 0 2 π d θ ′ δ 4 ( u − e i θ ′ u ′ ) = 8 r δ 3 ( x − x ′ ) (83) and we define
G ω ( x , x ′ ) = 1 8 ∫ 0 2 π d θ ′ g ( u , e i θ ′ u ′ ) . (84) G_\omega(x,x')=\frac{1}{8}\int_{0}^{2\pi}d\theta'\,g(u,e^{i\theta'}u'). \tag*{(84)} G ω ( x , x ′ ) = 8 1 ∫ 0 2 π d θ ′ g ( u , e i θ ′ u ′ ) . (84) Since h ^ \hat h h ^ g ( e i α u , e i θ ′ u ′ ) = g ( u , e i ( θ ′ − α ) u ′ ) g(e^{i\alpha}u,e^{i\theta'}u')=g(u,e^{i(\theta'-\alpha)}u') g ( e i α u , e i θ ′ u ′ ) = g ( u , e i ( θ ′ − α ) u ′ ) θ ′ \theta' θ ′ α \alpha α u u u x x x x ′ x' x ′ ω 2 ( h ^ − 4 M ) \omega^2(\hat h-4M) ω 2 ( h ^ − 4 M ) ω 2 ( h ^ − 4 M ) G ω = r δ 3 ( x − x ′ ) \omega^2(\hat h-4M)G_\omega=r\delta^3(x-x') ω 2 ( h ^ − 4 M ) G ω = r δ 3 ( x − x ′ ) (34) is r H ^ ω G ω = r δ 3 ( x − x ′ ) r\hat H_\omega G_\omega=r\delta^3(x-x') r H ^ ω G ω = r δ 3 ( x − x ′ ) H ^ ω G ω = δ 3 ( x − x ′ ) \hat H_\omega G_\omega=\delta^3(x-x') H ^ ω G ω = δ 3 ( x − x ′ ) r ′ r' r ′ (30) has been absorbed by the measure. In terms of paths, the θ ′ \theta' θ ′ x ′ x' x ′
For fixed t ~ \tilde t t ~ θ ′ \theta' θ ′ (37) runs from u ′ ′ ≡ e i θ ′ u ′ u''\equiv e^{i\theta'}u' u ′′ ≡ e i θ ′ u ′ u u u u ˙ 2 + u 2 \dot u^2+u^2 u ˙ 2 + u 2 u ¨ = u \ddot u=u u ¨ = u
u c l ( t ~ ′ ) = u ′ ′ sinh ( t ~ − t ~ ′ ) + u sinh t ~ ′ sinh t ~ (85) u_{\mathrm{cl}}(\tilde t')=\frac{u''\sinh(\tilde t-\tilde t')+u\sinh\tilde t'}{\sinh\tilde t} \tag*{(85)} u cl ( t ~ ′ ) = sinh t ~ u ′′ sinh ( t ~ − t ~ ′ ) + u sinh t ~ ′ (85) unique for every t ~ > 0 \tilde t>0 t ~ > 0 sinh t ~ ≠ 0 \sinh\tilde t\ne0 sinh t ~ = 0
A c l ( t ~ , θ ′ ) = ∫ 0 t ~ d t ~ ′ ( u ˙ c l 2 + u c l 2 ) = [ u c l ⋅ u ˙ c l ] 0 t ~ = ( r + r ′ ) coth t ~ − 2 u ⋅ u ′ ′ sinh t ~ (86) A_{\mathrm{cl}}(\tilde t,\theta')=\int_{0}^{\tilde t}d\tilde t'\,(\dot u_{\mathrm{cl}}^2+u_{\mathrm{cl}}^2)=[u_{\mathrm{cl}}\cdot\dot u_{\mathrm{cl}}]_{0}^{\tilde t}=(r+r')\coth\tilde t-\frac{2u\cdot u''}{\sinh\tilde t} \tag*{(86)} A cl ( t ~ , θ ′ ) = ∫ 0 t ~ d t ~ ′ ( u ˙ cl 2 + u cl 2 ) = [ u cl ⋅ u ˙ cl ] 0 t ~ = ( r + r ′ ) coth t ~ − sinh t ~ 2 u ⋅ u ′′ (86) with ∣ u ∣ 2 = r |u|^2=r ∣ u ∣ 2 = r ∣ u ′ ′ ∣ 2 = r ′ |u''|^2=r' ∣ u ′′ ∣ 2 = r ′ u = u c l + δ u u=u_{\mathrm{cl}}+\delta u u = u cl + δ u δ u ( 0 ) = δ u ( t ~ ) = 0 \delta u(0)=\delta u(\tilde t)=0 δ u ( 0 ) = δ u ( t ~ ) = 0 2 [ u ˙ c l ⋅ δ u ] 0 t ~ − 2 ∫ d t ~ ′ δ u ⋅ ( u ¨ c l − u c l ) 2[\dot u_{\mathrm{cl}}\cdot\delta u]_{0}^{\tilde t}-2\int d\tilde t'\,\delta u\cdot(\ddot u_{\mathrm{cl}}-u_{\mathrm{cl}}) 2 [ u ˙ cl ⋅ δ u ] 0 t ~ − 2 ∫ d t ~ ′ δ u ⋅ ( u ¨ cl − u cl ) e i ω A c l e^{i\omega A_{\mathrm{cl}}} e iω A cl t ~ \tilde t t ~ ( i / ω ) ∂ t ~ K = h ^ K (i/\omega)\partial_{\tilde t}K=\hat hK ( i / ω ) ∂ t ~ K = h ^ K K 1 = a ( t ~ ) e i ω Λ K_1=a(\tilde t)e^{i\omega\Lambda} K 1 = a ( t ~ ) e iω Λ Λ = ( u 2 + u ′ ′ 2 ) c − 2 u u ′ ′ / sh \Lambda=(u^2+u''^2)c-2uu''/\operatorname{sh} Λ = ( u 2 + u ′′2 ) c − 2 u u ′′ / sh c = coth t ~ c=\coth\tilde t c = coth t ~ sh = sinh t ~ \operatorname{sh}=\sinh\tilde t sh = sinh t ~ h ^ K 1 = [ − i 2 ω c + ( u c − u ′ ′ / sh ) 2 − u 2 ] K 1 \hat hK_1=[-\frac{i}{2\omega}c+(uc-u''/\operatorname{sh})^2-u^2]K_1 h ^ K 1 = [ − 2 ω i c + ( u c − u ′′ / sh ) 2 − u 2 ] K 1 ( i / ω ) ∂ t ~ K 1 = [ i ω a ′ / a + ( u 2 + u ′ ′ 2 − 2 u u ′ ′ cosh t ~ ) / sh 2 ] K 1 (i/\omega)\partial_{\tilde t}K_1=[\frac{i}{\omega}a'/a+(u^2+u''^2-2uu''\cosh\tilde t)/\operatorname{sh}^2]K_1 ( i / ω ) ∂ t ~ K 1 = [ ω i a ′ / a + ( u 2 + u ′′2 − 2 u u ′′ cosh t ~ ) / sh 2 ] K 1 c 2 − 1 = sh − 2 c^2-1=\operatorname{sh}^{-2} c 2 − 1 = sh − 2 a ′ / a = − 1 2 coth t ~ a'/a=-\frac{1}{2}\coth\tilde t a ′ / a = − 2 1 coth t ~ a ∝ sh − 1 / 2 a\propto\operatorname{sh}^{-1/2} a ∝ sh − 1/2 t ~ → 0 \tilde t\to0 t ~ → 0 K 1 K_1 K 1 p ^ 2 / 4 \hat p^2/4 p ^ 2 /4 ( ω / π i t ~ ) 1 / 2 e i ω ( u − u ′ ′ ) 2 / t ~ (\omega/\pi i\tilde t)^{1/2}e^{i\omega(u-u'')^2/\tilde t} ( ω / π i t ~ ) 1/2 e iω ( u − u ′′ ) 2 / t ~ a = ( ω / π i sh ) 1 / 2 a=(\omega/\pi i\operatorname{sh})^{1/2} a = ( ω / π i sh ) 1/2
∫ u ( t ~ ) = u u ( 0 ) = u ′ ′ D u e i ω ∫ 0 t ~ d t ~ ′ ( u ˙ 2 + u 2 ) = ⟨ u ∣ e − i ω t ~ h ^ ∣ u ′ ′ ⟩ = − ω 2 π 2 sinh 2 t ~ e i ω A c l ( t ~ , θ ′ ) (87) \int_{\substack{u(\tilde t)=u\\u(0)=u''}}\mathcal{D}u\,e^{i\omega\int_{0}^{\tilde t}d\tilde t'\,(\dot u^2+u^2)}
=\langle u|e^{-i\omega\tilde t\hat h}|u''\rangle
=-\frac{\omega^2}{\pi^2\sinh^2\tilde t}e^{i\omega A_{\mathrm{cl}}(\tilde t,\theta')} \tag*{(87)} ∫ u ( t ~ ) = u u ( 0 ) = u ′′ D u e iω ∫ 0 t ~ d t ~ ′ ( u ˙ 2 + u 2 ) = ⟨ u ∣ e − iω t ~ h ^ ∣ u ′′ ⟩ = − π 2 sinh 2 t ~ ω 2 e iω A cl ( t ~ , θ ′ ) (87) The terms of order 1 / ω 1/\omega 1/ ω ω \omega ω t ~ \tilde t t ~ θ ′ \theta' θ ′ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) [ 0 , 2 π ) [0,2\pi) [ 0 , 2 π ) § 4.1 . This is the difference with the Schwinger gauge, where keeping the classical path is the WKB approximation and fixing the proper time produces the diffraction integral [14 ] .
Differentiating Eq. (A.9) gives the Hamilton–Jacobi relations (42) . On the one hand − ∂ t ~ A c l = ( r + r ′ − 2 u ⋅ u ′ ′ cosh t ~ ) / sinh 2 t ~ -\partial_{\tilde{t}}\mathcal{A}_{\mathrm{cl}}=(r+r^{\prime}-2\boldsymbol{u}\cdot\boldsymbol{u}^{\prime\prime}\cosh\tilde{t})/\sinh^{2}\tilde{t} − ∂ t ~ A cl = ( r + r ′ − 2 u ⋅ u ′′ cosh t ~ ) / sinh 2 t ~ u ˙ c l ( 0 ) = ( u − u ′ ′ cosh t ~ ) / sinh t ~ \dot{\boldsymbol{u}}_{\mathrm{cl}}(0)=(\boldsymbol{u}-\boldsymbol{u}^{\prime\prime}\cosh\tilde{t})/\sinh\tilde{t} u ˙ cl ( 0 ) = ( u − u ′′ cosh t ~ ) / sinh t ~ E o s c = u ˙ c l 2 − u c l 2 E_{\mathrm{osc}}=\dot{\boldsymbol{u}}_{\mathrm{cl}}^{2}-\boldsymbol{u}_{\mathrm{cl}}^{2} E osc = u ˙ cl 2 − u cl 2 t ~ ′ = 0 \tilde{t}^{\prime}=0 t ~ ′ = 0 z ′ ′ = e i θ ′ z ′ z^{\prime\prime}=e^{i\theta^{\prime}}z^{\prime} z ′′ = e i θ ′ z ′ ∂ θ ′ ( u ⋅ u ′ ′ ) = − Im ( z † z ′ ′ ) \partial_{\theta^{\prime}}(\boldsymbol{u}\cdot\boldsymbol{u}^{\prime\prime})=-\operatorname{Im}(z^{\dagger}z^{\prime\prime}) ∂ θ ′ ( u ⋅ u ′′ ) = − Im ( z † z ′′ ) J = 2 Im ( z ′ ′ † z ˙ c l ( 0 ) ) = − 2 Im ( z † z ′ ′ ) / sinh t ~ J=2\operatorname{Im}(z^{\prime\prime\dagger}\dot{z}_{\mathrm{cl}}(0))=-2\operatorname{Im}(z^{\dagger}z^{\prime\prime})/\sinh\tilde{t} J = 2 Im ( z ′′ † z ˙ cl ( 0 )) = − 2 Im ( z † z ′′ ) / sinh t ~ ∂ θ ′ A c l = − J \partial_{\theta^{\prime}}\mathcal{A}_{\mathrm{cl}}=-J ∂ θ ′ A cl = − J u ⋅ u ′ ′ \boldsymbol{u}\cdot\boldsymbol{u}^{\prime\prime} u ⋅ u ′′ Re ( z † z ′ ′ ) \operatorname{Re}(z^{\dagger}z^{\prime\prime}) Re ( z † z ′′ )
u ⋅ u ′ ′ = Re ( e i θ ′ z † z ′ ) = ϖ cos ( θ ′ + γ ) , γ = arg ( z † z ′ ) , (88) \boldsymbol{u}\cdot\boldsymbol{u}^{\prime\prime}=\operatorname{Re}\left(e^{i\theta^{\prime}}z^{\dagger}z^{\prime}\right)=\varpi\cos(\theta^{\prime}+\gamma),\qquad\gamma=\arg(z^{\dagger}z^{\prime}), \tag*{(88)} u ⋅ u ′′ = Re ( e i θ ′ z † z ′ ) = ϖ cos ( θ ′ + γ ) , γ = arg ( z † z ′ ) , (88) and the θ ′ \theta^{\prime} θ ′
∫ 0 2 π d θ ′ exp [ − 2 i ω ϖ cos ( θ ′ + γ ) sinh t ~ ] = 2 π J 0 ( 2 ω ϖ sinh t ~ ) . (89) \int_{0}^{2\pi}d\theta^{\prime}\exp\left[-\frac{2i\omega\varpi\cos(\theta^{\prime}+\gamma)}{\sinh\tilde{t}}\right]=2\pi J_{0}\left(\frac{2\omega\varpi}{\sinh\tilde{t}}\right). \tag*{(89)} ∫ 0 2 π d θ ′ exp [ − sinh t ~ 2 iω ϖ cos ( θ ′ + γ ) ] = 2 π J 0 ( sinh t ~ 2 ω ϖ ) . (89) Inserting Eqs. (A.10) and (A.12) in Eq. (37) , the constants combine as i 8 ω ⋅ 2 π ⋅ ( − ω 2 π 2 ) = − i ω 4 π \frac{i}{8\omega}\cdot2\pi\cdot\left(-\frac{\omega^{2}}{\pi^{2}}\right)=-\frac{i\omega}{4\pi} 8 ω i ⋅ 2 π ⋅ ( − π 2 ω 2 ) = − 4 π iω (40) .
The far field and the lens plane
For an observer at x = r n ^ \boldsymbol{x}=r\hat{\boldsymbol{n}} x = r n ^
ϖ 2 = 1 2 ( r r ′ + x ⋅ x ′ ) = 1 2 r ( r ′ + n ^ ⋅ x ′ ) = r η 2 , (90) \varpi^{2}=\frac{1}{2}(rr^{\prime}+\boldsymbol{x}\cdot\boldsymbol{x}^{\prime})=\frac{1}{2}r(r^{\prime}+\hat{\boldsymbol{n}}\cdot\boldsymbol{x}^{\prime})=\frac{r\eta}{2}, \tag*{(90)} ϖ 2 = 2 1 ( r r ′ + x ⋅ x ′ ) = 2 1 r ( r ′ + n ^ ⋅ x ′ ) = 2 r η , (90) with η \eta η (46) . For large r r r ω ( r + r ′ ) coth t ~ \omega(r+r^{\prime})\coth\tilde{t} ω ( r + r ′ ) coth t ~ (40) varies rapidly unless coth t ~ − 1 ≃ 2 e − 2 t ~ \coth\tilde{t}-1\simeq2e^{-2\tilde{t}} coth t ~ − 1 ≃ 2 e − 2 t ~ 1 / ω r 1/\omega r 1/ ω r t ~ = χ + 1 2 ln ( 2 ω r ) \tilde{t}=\chi+\frac{1}{2}\ln(2\omega r) t ~ = χ + 2 1 ln ( 2 ω r ) q ≡ e − 2 t ~ = e − 2 χ / 2 ω r q\equiv e^{-2\tilde{t}}=e^{-2\chi}/2\omega r q ≡ e − 2 t ~ = e − 2 χ /2 ω r χ \chi χ r → ∞ r\to\infty r → ∞ χ \chi χ d t ~ = d l / 2 r d\tilde{t}=dl/2r d t ~ = d l /2 r χ \chi χ sinh t ~ = e t ~ ( 1 − q ) / 2 \sinh\tilde{t}=e^{\tilde{t}}(1-q)/2 sinh t ~ = e t ~ ( 1 − q ) /2 q = O ( 1 / ω r ) q=O(1/\omega r) q = O ( 1/ ω r ) (40) become
1 sinh 2 t ~ = 4 q ( 1 − q ) 2 = 2 e − 2 χ ω r [ 1 + O ( 1 ω r ) ] , e i ω t ~ = e i ω χ e i ν ln ( 2 ω r ) , \frac{1}{\sinh^{2}\tilde{t}}=\frac{4q}{(1-q)^{2}}=\frac{2e^{-2\chi}}{\omega r}\left[1+O\left(\frac{1}{\omega r}\right)\right],\qquad e^{i\omega\tilde{t}}=e^{i\omega\chi}e^{i\nu\ln(2\omega r)}, sinh 2 t ~ 1 = ( 1 − q ) 2 4 q = ω r 2 e − 2 χ [ 1 + O ( ω r 1 ) ] , e iω t ~ = e iω χ e i ν l n ( 2 ω r ) , ω ( r + r ′ ) coth t ~ = ω ( r + r ′ ) 1 + q 1 − q = ω r + ω r ′ + e − 2 χ + O ( r ′ r , 1 ω r ) , \omega(r+r^{\prime})\coth\tilde{t}=\omega(r+r^{\prime})\frac{1+q}{1-q}=\omega r+\omega r^{\prime}+e^{-2\chi}+O\left(\frac{r^{\prime}}{r},\frac{1}{\omega r}\right), ω ( r + r ′ ) coth t ~ = ω ( r + r ′ ) 1 − q 1 + q = ω r + ω r ′ + e − 2 χ + O ( r r ′ , ω r 1 ) , 2 ω ϖ sinh t ~ = 4 ω ϖ q 1 / 2 1 − q = 2 ζ e − χ [ 1 + O ( 1 ω r ) ] . (91) \frac{2\omega\varpi}{\sinh\tilde{t}}=\frac{4\omega\varpi q^{1/2}}{1-q}=2\sqrt{\zeta}e^{-\chi}\left[1+O\left(\frac{1}{\omega r}\right)\right]. \tag*{(91)} sinh t ~ 2 ω ϖ = 1 − q 4 ω ϖ q 1/2 = 2 ζ e − χ [ 1 + O ( ω r 1 ) ] . (91) with ν = ω / 2 \nu=\omega/2 ν = ω /2 ζ = ω η \zeta=\omega\eta ζ = ω η
G ω ( r n ^ , x ′ ) → − i 2 π r e i ω r + i ν ln 2 ω r e i ω r ′ ∫ − ∞ ∞ d χ e − 2 χ e i ( ω χ + e − 2 χ ) J 0 ( 2 ζ e − χ ) . (92) G_{\omega}(r\hat{\boldsymbol{n}},\boldsymbol{x}^{\prime})\to-\frac{i}{2\pi r}e^{i\omega r+i\nu\ln2\omega r}e^{i\omega r^{\prime}}\int_{-\infty}^{\infty}d\chi\,e^{-2\chi}e^{i(\omega\chi+e^{-2\chi})}J_{0}(2\sqrt{\zeta}e^{-\chi}). \tag*{(92)} G ω ( r n ^ , x ′ ) → − 2 π r i e iω r + i ν l n 2 ω r e iω r ′ ∫ − ∞ ∞ d χ e − 2 χ e i ( ω χ + e − 2 χ ) J 0 ( 2 ζ e − χ ) . (92) Writing J 0 J_{0} J 0 J 0 ( z ) = 1 2 π ∫ 0 2 π d θ e i z cos θ J_{0}(z)=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,e^{iz\cos\theta} J 0 ( z ) = 2 π 1 ∫ 0 2 π d θ e i z c o s θ θ = θ ′ + γ + π \theta=\theta^{\prime}+\gamma+\pi θ = θ ′ + γ + π e i ω r ′ = e i ζ e − i ω n ^ ⋅ x ′ e^{i\omega r^{\prime}}=e^{i\zeta}e^{-i\omega\hat{\boldsymbol{n}}\cdot\boldsymbol{x}^{\prime}} e iω r ′ = e i ζ e − iω n ^ ⋅ x ′ (47) and (48) . The only limit taken is r → ∞ r\to\infty r → ∞ (40) and (47) is 7 × 10 − 4 7\times10^{-4} 7 × 1 0 − 4 ω r = 10 4 \omega r=10^{4} ω r = 1 0 4 2.5 × 10 − 4 2.5\times10^{-4} 2.5 × 1 0 − 4 ω r = 3 × 10 4 \omega r=3\times10^{4} ω r = 3 × 1 0 4 ω ≃ 2 \omega\simeq2 ω ≃ 2 ω r ′ ≃ 4.8 \omega r^{\prime}\simeq4.8 ω r ′ ≃ 4.8
After the fibre integral, Eq. (47) reads e − i ζ F ω = − 2 i ∫ d χ e i ω χ e − 2 χ e i e − 2 χ J 0 ( 2 ζ e − χ ) e^{-i\zeta}F_{\omega}=-2i\int d\chi\,e^{i\omega\chi}e^{-2\chi}e^{ie^{-2\chi}}J_{0}(2\sqrt{\zeta}e^{-\chi}) e − i ζ F ω = − 2 i ∫ d χ e iω χ e − 2 χ e i e − 2 χ J 0 ( 2 ζ e − χ ) M = 0 M=0 M = 0 s = e − 2 χ s=e^{-2\chi} s = e − 2 χ ∫ 0 ∞ d s e − p s J 0 ( 2 ζ s ) = p − 1 e − ζ / p \int_{0}^{\infty}ds\,e^{-ps}J_{0}(2\sqrt{\zeta s})=p^{-1}e^{-\zeta/p} ∫ 0 ∞ d s e − p s J 0 ( 2 ζ s ) = p − 1 e − ζ / p e − i ζ F ω = e − i ζ e^{-i\zeta}F_\omega=e^{-i\zeta} e − i ζ F ω = e − i ζ F ω = 1 F_\omega=1 F ω = 1 M ≠ 0 M\ne0 M = 0 v = e − χ v=e^{-\chi} v = e − χ − 2 i ∫ 0 ∞ d v v 1 − i w e i v 2 J 0 ( 2 ζ v ) -2i\int_0^\infty dv\,v^{1-iw}e^{iv^2}J_0(2\sqrt{\zeta}\,v) − 2 i ∫ 0 ∞ d v v 1 − i w e i v 2 J 0 ( 2 ζ v ) [74 ]
∫ 0 ∞ v p − 1 e − α v 2 J 0 ( c v ) d v = Γ ( p / 2 ) 2 α p / 2 1 F 1 ( p 2 ; 1 ; − c 2 4 α ) (93) \int_0^\infty v^{p-1}e^{-\alpha v^2}J_0(cv)\,dv=\frac{\Gamma(p/2)}{2\alpha^{p/2}}{}_1F_1\left(\frac{p}{2};1;-\frac{c^2}{4\alpha}\right) \tag*{(93)} ∫ 0 ∞ v p − 1 e − α v 2 J 0 ( c v ) d v = 2 α p /2 Γ ( p /2 ) 1 F 1 ( 2 p ; 1 ; − 4 α c 2 ) (93) with p = 2 − i w p=2-iw p = 2 − i w α = − i + 0 \alpha=-i+0 α = − i + 0 c = 2 ζ c=2\sqrt{\zeta} c = 2 ζ − c 2 / 4 α = − i ζ -c^2/4\alpha=-i\zeta − c 2 /4 α = − i ζ − i ( − i ) − 1 = 1 -i(-i)^{-1}=1 − i ( − i ) − 1 = 1 ( − i ) i ν = e π ν / 2 (-i)^{i\nu}=e^{\pi\nu/2} ( − i ) i ν = e π ν /2 e − i ζ F ω = e π ν / 2 Γ ( 1 − i ν ) 1 F 1 ( 1 − i ν ; 1 ; − i ζ ) e^{-i\zeta}F_\omega=e^{\pi\nu/2}\Gamma(1-i\nu){}_1F_1(1-i\nu;1;-i\zeta) e − i ζ F ω = e π ν /2 Γ ( 1 − i ν ) 1 F 1 ( 1 − i ν ; 1 ; − i ζ ) 1 F 1 ( a ; b ; x ) = e x 1 F 1 ( b − a ; b ; − x ) {}_1F_1(a;b;x)=e^x{}_1F_1(b-a;b;-x) 1 F 1 ( a ; b ; x ) = e x 1 F 1 ( b − a ; b ; − x ) e − i ζ e^{-i\zeta} e − i ζ
F ω ( x ′ ) = N w 1 F 1 ( i ν , 1 ; i w η ) , N w = e π ν / 2 Γ ( 1 − i ν ) , (94) F_\omega(x')=N_w{}_1F_1(i\nu,1;iw\eta),\qquad N_w=e^{\pi\nu/2}\Gamma(1-i\nu), \tag*{(94)} F ω ( x ′ ) = N w 1 F 1 ( i ν , 1 ; i w η ) , N w = e π ν /2 Γ ( 1 − i ν ) , (94) in agreement with Eq. (49) . Evaluated with 40-digit arithmetic, the integral agrees with this closed form to 3 × 10 − 37 3\times10^{-37} 3 × 1 0 − 37
With the variables of Eq. (50) , e − χ = w / 2 ∣ x ∣ e^{-\chi}=\sqrt{w/2}\,|x| e − χ = w /2 ∣ x ∣ ζ = w / 2 y \sqrt{\zeta}=\sqrt{w/2}\,y ζ = w /2 y x ⋅ y = − ∣ x ∣ y cos θ \boldsymbol{x}\cdot\boldsymbol{y}=-|x|y\cos\theta x ⋅ y = − ∣ x ∣ y cos θ (48) become e − 2 χ = w 2 ∣ x ∣ 2 e^{-2\chi}=\frac{w}{2}|x|^2 e − 2 χ = 2 w ∣ x ∣ 2 e − χ cos θ = − w x ⋅ y e^{-\chi}\cos\theta=-w\boldsymbol{x}\cdot\boldsymbol{y} e − χ cos θ = − w x ⋅ y w χ = − w ln ∣ x ∣ − ν ln ( w / 2 ) w\chi=-w\ln|x|-\nu\ln(w/2) w χ = − w ln ∣ x ∣ − ν ln ( w /2 ) w 2 y 2 = ζ \frac{w}{2}y^2=\zeta 2 w y 2 = ζ
Ψ = w [ 1 2 ∣ x ∣ 2 − x ⋅ y − ln ∣ x ∣ ] − ν ln w 2 = w T ( x , y ) − ζ − ν ln w 2 . (95) \Psi=w\left[\frac{1}{2}|x|^2-\boldsymbol{x}\cdot\boldsymbol{y}-\ln|x|\right]-\nu\ln\frac{w}{2}=wT(\boldsymbol{x},\boldsymbol{y})-\zeta-\nu\ln\frac{w}{2}. \tag*{(95)} Ψ = w [ 2 1 ∣ x ∣ 2 − x ⋅ y − ln ∣ x ∣ ] − ν ln 2 w = w T ( x , y ) − ζ − ν ln 2 w . (95) For the measure, d 2 x = ∣ x ∣ d ∣ x ∣ d θ d^2x=|x|\,d|x|\,d\theta d 2 x = ∣ x ∣ d ∣ x ∣ d θ ∣ x ∣ d ∣ x ∣ = − 2 w e − 2 χ d χ |x|\,d|x|=-\frac{2}{w}e^{-2\chi}d\chi ∣ x ∣ d ∣ x ∣ = − w 2 e − 2 χ d χ χ \chi χ − ∞ -\infty − ∞ ∞ \infty ∞ ∣ x ∣ |x| ∣ x ∣ ∞ \infty ∞ 0 0 0 e − 2 χ d χ d θ = w 2 d 2 x e^{-2\chi}d\chi\,d\theta=\frac{w}{2}d^2x e − 2 χ d χ d θ = 2 w d 2 x ( χ , θ ) (\chi,\theta) ( χ , θ ) (47) ,
e i ν ln ( w / 2 ) F ω = w 2 π i ∫ d 2 x e i w T ( x , y ) , (96) e^{i\nu\ln(w/2)}F_\omega=\frac{w}{2\pi i}\int d^2x\,e^{iwT(\boldsymbol{x},\boldsymbol{y})}, \tag*{(96)} e i ν l n ( w /2 ) F ω = 2 π i w ∫ d 2 x e i w T ( x , y ) , (96) which has the form of the point-lens integral (12) for F ( w , y ) ≡ e i ν ln ( w / 2 ) F ω F(w,y)\equiv e^{i\nu\ln(w/2)}F_\omega F ( w , y ) ≡ e i ν l n ( w /2 ) F ω ( − i ) i ν = e π ν / 2 (-i)^{i\nu}=e^{\pi\nu/2} ( − i ) i ν = e π ν /2 (13) to Eq. (B.5) is ( − i w / 2 ) i ν e − π ν / 2 = e i ν ln ( w / 2 ) (-iw/2)^{i\nu}e^{-\pi\nu/2}=e^{i\nu\ln(w/2)} ( − i w /2 ) i ν e − π ν /2 = e i ν l n ( w /2 ) y 2 = η / 2 M y^2=\eta/2M y 2 = η /2 M η = r ′ ( 1 − cos β ) ≃ r ′ β 2 / 2 \eta=r'(1-\cos\beta)\simeq r'\beta^2/2 η = r ′ ( 1 − cos β ) ≃ r ′ β 2 /2 y ≃ r ′ β / ξ E y\simeq r'\beta/\xi_E y ≃ r ′ β / ξ E ξ E = 4 M r ′ \xi_E=\sqrt{4Mr'} ξ E = 4 M r ′
Near an image the integral (47) is Gaussian. Its amplitude is the prefactor 1 / π 1/\pi 1/ π e − 2 χ = w 2 x 2 e^{-2\chi}=\frac{w}{2}x^2 e − 2 χ = 2 w x 2 2 π ∣ det ∂ 2 Ψ ∣ − 1 / 2 = 2 π [ w 2 ( 1 + x 2 ) ∣ x y ∣ ] − 1 / 2 2\pi|\det\partial^2\Psi|^{-1/2}=2\pi[w^2(1+x^2)|xy|]^{-1/2} 2 π ∣ det ∂ 2 Ψ ∣ − 1/2 = 2 π [ w 2 ( 1 + x 2 ) ∣ x y ∣ ] − 1/2 ∣ x y ∣ = ∣ x 2 − 1 ∣ |xy|=|x^2-1| ∣ x y ∣ = ∣ x 2 − 1∣ ( 1 + x 2 ) ∣ x y ∣ = ∣ x 4 − 1 ∣ (1+x^2)|xy|=|x^4-1| ( 1 + x 2 ) ∣ x y ∣ = ∣ x 4 − 1∣
x 2 ∣ x 4 − 1 ∣ = ∣ 1 − x − 4 ∣ − 1 / 2 = μ ± . (97) \frac{x^2}{\sqrt{|x^4-1|}}=|1-x^{-4}|^{-1/2}=\sqrt{\mu_\pm}. \tag*{(97)} ∣ x 4 − 1∣ x 2 = ∣1 − x − 4 ∣ − 1/2 = μ ± . (97) Each positive curvature contributes a phase e i π / 4 e^{i\pi/4} e iπ /4 e i π / 2 e^{i\pi/2} e iπ /2 − i -i − i − i -i − i
